Greatest common divisor

From Rosetta Code


Task
Greatest common divisor
You are encouraged to solve this task according to the task description, using any language you may know.
Task

Find the greatest common divisor   (GCD)   of two integers.


Greatest common divisor   is also known as   greatest common factor (gcf)   and   greatest common measure.


Related task


See also



11l[edit]

Translation of: Python
F gcd(=u, =v)
   L v != 0
      (u, v) = (v, u % v)
   R abs(u)

print(gcd(0, 0))
print(gcd(0, 10))
print(gcd(0, -10))
print(gcd(9, 6))
print(gcd(6, 9))
print(gcd(-6, 9))
print(gcd(8, 45))
print(gcd(40902, 24140))
Output:
0
10
10
3
3
3
1
34

360 Assembly[edit]

Translation of: FORTRAN

For maximum compatibility, this program uses only the basic instruction set (S/360) with 2 ASSIST macros (XDECO,XPRNT).

*        Greatest common divisor   04/05/2016
GCD      CSECT
         USING  GCD,R15            use calling register
         L      R6,A               u=a
         L      R7,B               v=b
LOOPW    LTR    R7,R7              while v<>0 
         BZ     ELOOPW               leave while
         LR     R8,R6                t=u
         LR     R6,R7                u=v
         LR     R4,R8                t
         SRDA   R4,32                shift to next reg
         DR     R4,R7                t/v
         LR     R7,R4                v=mod(t,v)
         B      LOOPW              end while
ELOOPW   LPR    R9,R6              c=abs(u)
         L      R1,A               a	
         XDECO  R1,XDEC            edit a
         MVC    PG+4(5),XDEC+7     move a to buffer
         L      R1,B               b
         XDECO  R1,XDEC            edit b
         MVC    PG+10(5),XDEC+7    move b to buffer
         XDECO  R9,XDEC            edit c
         MVC    PG+17(5),XDEC+7    move c to buffer
         XPRNT  PG,80              print buffer
         XR     R15,R15            return code =0
         BR     R14                return to caller
A        DC     F'1071'            a
B        DC     F'1029'            b
PG       DC     CL80'gcd(00000,00000)=00000'  buffer
XDEC     DS     CL12               temp for edit
         YREGS
         END    GCD
Output:
gcd( 1071, 1029)=   21

8th[edit]

: gcd \ a b -- gcd
	dup 0 n:= if drop ;; then
	tuck \ b a b
	n:mod \ b a-mod-b
	recurse ; 
	
: demo \ a b --
	2dup "GCD of " . . " and " . . " = " . gcd . ;

100    5 demo cr
  5  100 demo cr
  7   23 demo cr

bye
Output:
GCD of 5 and 100 = 5
GCD of 100 and 5 = 5
GCD of 23 and 7 = 1

AArch64 Assembly[edit]

Works with: as version Raspberry Pi 3B version Buster 64 bits
/* ARM assembly AARCH64 Raspberry PI 3B */
/*  program calPgcd64.s  */
 
/*******************************************/
/* Constantes file                         */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"

/*********************************/
/* Initialized data              */
/*********************************/
.data
sMessResult:        .asciz "Number 1 : @ number 2 : @ PGCD  : @ \n"
szCarriageReturn:   .asciz "\n"
szMessError:        .asciz "Error PGCD !!\n"

/*********************************/
/* UnInitialized data            */
/*********************************/
.bss
sZoneConv:            .skip 24
/*********************************/
/*  code section                 */
/*********************************/
.text
.global main 
main:                               // entry of program 

    mov x20,36
    mov x21,18
    mov x0,x20
    mov x1,x21
    bl calPGCDmod
    bcs   99f                       // error ?
    mov x2,x0                       // pgcd
    mov x0,x20
    mov x1,x21
    bl displayResult
    mov x20,37
    mov x21,15
    mov x0,x20
    mov x1,x21
    bl calPGCDmod
    bcs   99f                       // error ?
    mov x2,x0                       // pgcd
    mov x0,x20
    mov x1,x21
    bl displayResult
    
    
    b 100f
99:                                 // display error
    ldr x0,qAdrszMessError
    bl affichageMess
100:                                // standard end of the program 
    mov x0, #0                      // return code
    mov x8, #EXIT                   // request to exit program
    svc #0                          // perform the system call
 
qAdrszCarriageReturn:     .quad szCarriageReturn
qAdrszMessError:          .quad szMessError

/***************************************************/
/*   Compute pgcd  modulo use */
/***************************************************/
/* x0 contains first number */
/* x1 contains second number */
/* x0 return  PGCD            */
/* if error carry set to 1    */
calPGCDmod:
    stp x1,lr,[sp,-16]!        // save  registres
    stp x2,x3,[sp,-16]!        // save  registres
    cbz x0,99f                 // if = 0 error
    cbz x1,99f
    cmp x0,0
    bgt 1f
    neg x0,x0                  // if negative inversion number 1
1:
    cmp x1,0
    bgt 2f
    neg x1,x1                  // if negative inversion number 2
2:
    cmp x0,x1                  // compare two numbers
    bgt 3f
    mov x2,x0                  // inversion
    mov x0,x1
    mov x1,x2
3:
    udiv x2,x0,x1              // division
    msub x0,x2,x1,x0           // compute remainder
    cmp x0,0
    bgt 2b                     // loop
    mov x0,x1
    cmn x0,0                   // clear carry
    b 100f
99:                            // error
    mov x0,0
    cmp x0,0                   // set carry
100:
    ldp x2,x3,[sp],16          // restaur des  2 registres
    ldp x1,lr,[sp],16          // restaur des  2 registres
    ret                        // retour adresse lr x30
    
/***************************************************/
/*   display result */
/***************************************************/
/* x0 contains first number */
/* x1 contains second number */
/* x2 contains  PGCD         */
displayResult:
    stp x1,lr,[sp,-16]!          // save  registres
    mov x3,x1                    // save x1
    ldr x1,qAdrsZoneConv
    bl conversion10              // décimal conversion 
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv         // insert conversion
    bl strInsertAtCharInc
    mov x4,x0                    // save message address
    mov x0,x3                    // conversion second number
    ldr x1,qAdrsZoneConv
    bl conversion10              // décimal conversion 
    mov x0,x4                    // move message address
    ldr x1,qAdrsZoneConv         // insert conversion
    bl strInsertAtCharInc
    mov x4,x0                    // save message address
    mov x0,x2                    // conversion pgcd
    ldr x1,qAdrsZoneConv
    bl conversion10              // décimal conversion 
    mov x0,x4                    // move message address
    ldr x1,qAdrsZoneConv         // insert conversion
    bl strInsertAtCharInc
    bl affichageMess             // display message
    ldp x1,lr,[sp],16            // restaur des  2 registres
    ret                          // retour adresse lr x30
qAdrsMessResult:          .quad sMessResult
qAdrsZoneConv:            .quad sZoneConv
/********************************************************/
/*        File Include fonctions                        */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"

ACL2[edit]

(include-book "arithmetic-3/floor-mod/floor-mod" :dir :system)

(defun gcd$ (x y)
   (declare (xargs :guard (and (natp x) (natp y))))
   (cond ((or (not (natp x)) (< y 0))
          nil)
         ((zp y) x)
         (t (gcd$ y (mod x y)))))

Action![edit]

CARD FUNC Gcd(CARD a,b)
  CARD tmp

  IF a<b THEN
    tmp=a a=b b=tmp
  FI

  WHILE b#0
  DO
    tmp=a MOD b
    a=b 
    b=tmp
  OD
RETURN(a)

PROC Test(CARD a,b)
  CARD res

  res=Gcd(a,b)
  PrintF("GCD of %I and %I is %I%E",a,b,res)
RETURN

PROC Main()
  Test(48,18)
  Test(9360,12240)
  Test(17,19)
  Test(123,1)
  Test(0,0)
RETURN
Output:

Screenshot from Atari 8-bit computer

GCD of 48 and 18 is 6
GCD of 9360 and 12240 is 720
GCD of 17 and 19 is 1
GCD of 123 and 1 is 1
GCD of 0 and 0 is 0

ActionScript[edit]

//Euclidean algorithm
function gcd(a:int,b:int):int
{
	var tmp:int;
	//Swap the numbers so a >= b
	if(a < b)
	{
		tmp = a;
		a = b;
		b = tmp;
	}
	//Find the gcd
	while(b != 0)
	{
		tmp = a % b;
		a = b;
		b = tmp;
	}
	return a;
}

Ada[edit]

with Ada.Text_Io; use Ada.Text_Io;

procedure Gcd_Test is
   function Gcd (A, B : Integer) return Integer is
      M : Integer := A;
      N : Integer := B;
      T : Integer;
   begin
      while N /= 0 loop
         T := M;
         M := N;
         N := T mod N;
      end loop;
      return M;
   end Gcd;
   
begin
   Put_Line("GCD of 100, 5 is" & Integer'Image(Gcd(100, 5)));
   Put_Line("GCD of 5, 100 is" & Integer'Image(Gcd(5, 100)));
   Put_Line("GCD of 7, 23 is" & Integer'Image(Gcd(7, 23)));
end Gcd_Test;

Output:

GCD of 100, 5 is 5
GCD of 5, 100 is 5
GCD of 7, 23 is 1

Aime[edit]

o_integer(gcd(33, 77));
o_byte('\n');
o_integer(gcd(49865, 69811));
o_byte('\n');

ALGOL 60[edit]

begin
    comment Greatest common divisor - algol 60;
	
    integer procedure gcd(m,n);
 	value m,n;
	integer m,n;
    begin
        integer a,b;
        a:=abs(m);
        b:=abs(n);
        if a=0 then gcd:=b
        else begin
			integer c,i;
		    for i:=a while b notequal 0 do begin
                c:=b;
                b:=a-(a div b)*b;
                a:=c
            end;
            gcd:=a
        end
    end gcd;
	
    outinteger(1,gcd(21,35))
end
Output:
 7


ALGOL 68[edit]

Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
PROC gcd = (INT a, b) INT: (
  IF a = 0 THEN
    b
  ELIF b = 0 THEN
    a
  ELIF a > b  THEN
    gcd(b, a MOD b)
  ELSE
    gcd(a, b MOD a)
  FI     
);
test:(
  INT a = 33, b = 77;
  printf(($x"The gcd of"g" and "g" is "gl$,a,b,gcd(a,b)));
  INT c = 49865, d = 69811;
  printf(($x"The gcd of"g" and "g" is "gl$,c,d,gcd(c,d)))
)

Output:

 The gcd of        +33 and         +77 is         +11
 The gcd of     +49865 and      +69811 is       +9973

ALGOL-M[edit]

BEGIN

%  RETURN P MOD Q  %
INTEGER FUNCTION MOD (P, Q);
INTEGER P, Q;
BEGIN
    MOD := P - Q * (P / Q);
END;

%  RETURN GREATEST COMMON DIVISOR OF X AND Y  %
INTEGER FUNCTION GCD (X, Y);
INTEGER X, Y;
BEGIN
	INTEGER R;
	IF X < Y THEN
	BEGIN
		INTEGER TEMP;
		TEMP := X;
		X := Y;
		Y := TEMP;
	END;
	WHILE (R := MOD(X, Y)) <> 0 DO
	BEGIN
		X := Y;
		Y := R;
	END;
	GCD := Y;
END;

COMMENT - EXERCISE THE FUNCTION;

WRITE("THE GDC OF 21 AND 35 IS", GCD(21,35));
WRITE("THE GDC OF 23 AND 35 IS", GCD(23,35));
WRITE("THE GDC OF 1071 AND 1029 IS", GCD(1071,1029));
WRITE("THE GDC OF 3528 AND 3780 IS", GCD(3528,252));

END
Output:
THE GDC OF 21 AND 35 IS    7
THE GDC OF 23 AND 35 IS    1
THE GDC OF 1071 AND 1029 IS   21
THE GDC OF 3528 AND 3780 IS  252

ALGOL W[edit]

begin
    % iterative Greatest Common Divisor routine                               %
    integer procedure gcd ( integer value m, n ) ;
    begin
        integer a, b, newA;
        a := abs( m );
        b := abs( n );
        if a = 0 then begin
            b
            end
        else begin
            while b not = 0 do begin
                newA := b;
                b    := a rem b;
                a    := newA;
            end;
            a
        end
    end gcd ;

    write( gcd( -21, 35 ) );
end.

Alore[edit]

def gcd(a as Int, b as Int) as Int
   while b != 0
      a,b = b, a mod b
   end
   return Abs(a)
end

AntLang[edit]

AntLang has a built-in gcd function.

gcd[33; 77]

It is not recommended, but possible to implement it on your own.

/Unoptimized version
gcd':{a:x;b:y;last[{(0 eq a mod x) min (0 eq b mod x)} hfilter {1 + x} map range[a max b]]}

APL[edit]

Works with: Dyalog APL
       33 49865 ∨ 77 69811 
11 9973

If you're interested in how you'd write GCD in Dyalog, if Dyalog didn't have a primitive for it, (i.e. using other algorithms mentioned on this page: iterative, recursive, binary recursive), see different ways to write GCD in Dyalog.

Works with: APL2
       ⌈/(^/0=A∘.|X)/A←⍳⌊/X←49865 69811 
9973


AppleScript[edit]

By recursion:

-- gcd :: Int -> Int -> Int
on gcd(a, b)
    if b  0 then
        gcd(b, a mod b)
    else
        if a < 0 then
            -a
        else
            a
        end if
    end if
end gcd

And just for the sake of it, the same thing iteratively:

on hcf(a, b)
    repeat until (b = 0)
        set x to a
        set a to b
        set b to x mod b
    end repeat
    
    if (a < 0) then return -a
    return a
end hcf

Arendelle[edit]

< a , b >

( r , @a )

[ @r != 0 ,

        ( r , @a % @b )

        { @r != 0 ,

                ( a , @b )
                ( b , @r )

        }
]

( return , @b )

Arturo[edit]

print gcd [10 15]
Output:
5

ATS[edit]

Works with: ATS version Postiats 0.4.1


Stein’s algorithm, without proofs[edit]

Here is an implementation of Stein’s algorithm, without proofs of termination or correctness.

(********************************************************************)
(*

  GCD of two integers, by Stein’s algorithm:
  https://en.wikipedia.org/w/index.php?title=Binary_GCD_algorithm&oldid=1072393147

  This is an implementation without proofs of anything.

  The implementations shown here require the GCC builtin functions
  for ‘count trailing zeros’. If your C compiler is GCC or another
  that supports those functions, you are fine. Otherwise, one could
  easily substitute other C code.

  Compile with ‘patscc -o gcd gcd.dats’.

*)

#define ATS_EXTERN_PREFIX "rosettacode_gcd_"
#define ATS_DYNLOADFLAG 0       (* No initialization is needed. *)

#include "share/atspre_define.hats"
#include "share/atspre_staload.hats"

(********************************************************************)
(*                                                                  *)
(* Declarations of the functions.                                   *)
(*                                                                  *)

(* g0uint_gcd_stein will be the generic template function for
   unsigned integers. *)
extern fun {tk : tkind}
g0uint_gcd_stein :
  (g0uint tk, g0uint tk) -<> g0uint tk

(* g0int_gcd_stein will be the generic template function for
   signed integers, giving an unsigned result. *)
extern fun {tk_signed, tk_unsigned : tkind}
g0int_gcd_stein :
  (g0int tk_signed, g0int tk_signed) -<> g0uint tk_unsigned

(* Let us call these functions ‘gcd_stein’ or just ‘gcd’. *)
overload gcd_stein with g0uint_gcd_stein
overload gcd_stein with g0int_gcd_stein
overload gcd with gcd_stein

(********************************************************************)
(*                                                                  *)
(* The implementations.                                             *)
(*                                                                  *)

%{^

/*

  We will need a ‘count trailing zeros of a positive number’ function,
  but this is not provided in the ATS prelude. Here are
  implementations using GCC builtin functions. For fast alternatives
  in standard C, see
  https://www.chessprogramming.org/index.php?title=BitScan&oldid=22495#Trailing_Zero_Count

*/

ATSinline() atstype_uint
rosettacode_gcd_g0uint_ctz_uint (atstype_uint x)
{
  return __builtin_ctz (x);
}

ATSinline() atstype_ulint
rosettacode_gcd_g0uint_ctz_ulint (atstype_ulint x)
{
  return __builtin_ctzl (x);
}

ATSinline() atstype_ullint
rosettacode_gcd_g0uint_ctz_ullint (atstype_ullint x)
{
  return __builtin_ctzll (x);
}

%}

extern fun g0uint_ctz_uint : uint -<> int = "mac#%"
extern fun g0uint_ctz_ulint : ulint -<> int = "mac#%"
extern fun g0uint_ctz_ullint : ullint -<> int = "mac#%"

(* A generic template function for ‘count trailing zeros’ of
   non-dependent unsigned integers. *)
extern fun {tk : tkind} g0uint_ctz : g0uint tk -<> int

(* Link the implementations to the template function. *)
implement g0uint_ctz<uint_kind> (x) = g0uint_ctz_uint x
implement g0uint_ctz<ulint_kind> (x) = g0uint_ctz_ulint x
implement g0uint_ctz<ullint_kind> (x) = g0uint_ctz_ullint x

(* Let one call the function simply ‘ctz’. *)
overload ctz with g0uint_ctz

(* Now the actual implementation of g0uint_gcd_stein, the template
   function for the gcd of two unsigned integers. *)
implement {tk}
g0uint_gcd_stein (u, v) =
  let
    (* Make ‘t’ a shorthand for the unsigned integer type. *)
    typedef t = g0uint tk

    (* Use this macro to fake proof that an int is non-negative. *)
    macdef nonneg (n) = $UNSAFE.cast{intGte 0} ,(n)

    (* Looping is done by tail recursion. There is no proof
       the function terminates; this fact is indicated by
       ‘<!ntm>’. *)
    fun {tk : tkind}
    main_loop (x_odd : t, y : t) :<!ntm> t =
      let
        (* Remove twos from y, giving an odd number.
           Note gcd(x_odd,y_odd) = gcd(x_odd,y). *)
        val y_odd = (y >> nonneg (ctz y))
      in
        if x_odd = y_odd then
          x_odd
        else
          let
            (* If y_odd < x_odd then swap x_odd and y_odd.
               This operation does not affect the gcd. *)
            val x_odd = min (x_odd, y_odd)
            and y_odd = max (x_odd, y_odd)
          in
            main_loop (x_odd, y_odd - x_odd)
          end
      end

    fn
    u_and_v_both_positive (u : t, v : t) :<> t =
      let
        (* n = the number of common factors of two in u and v. *)
        val n = ctz (u lor v)

        (* Remove the common twos from u and v, giving x and y. *)
        val x = (u >> nonneg n)
        val y = (v >> nonneg n)

        (* Remove twos from x, giving an odd number.
           Note gcd(x_odd,y) = gcd(x,y). *)
        val x_odd = (x >> nonneg (ctz x))

        (* Run the main loop, but pretend it is proven to
           terminate. Otherwise we could not write ‘<>’ above,
           telling the ATS compiler that we trust the function
           to terminate. *)
        val z = $effmask_ntm (main_loop (x_odd, y))
      in
        (* Put the common factors of two back in. *)
        (z << nonneg n)
      end

    (* If v < u then swap u and v. This operation does not
       affect the gcd. *)
    val u = min (u, v)
    and v = max (u, v)
  in
    if iseqz u then
      v
    else
      u_and_v_both_positive (u, v)
  end

(* The implementation of g0int_gcd_stein, the template function for
   the gcd of two signed integers, giving an unsigned result. *)
implement {signed_tk, unsigned_tk}
g0int_gcd_stein (u, v) =
  let
    val abs_u = $UNSAFE.cast{g0uint unsigned_tk} (abs u)
    val abs_v = $UNSAFE.cast{g0uint unsigned_tk} (abs v)
  in
    g0uint_gcd_stein<unsigned_tk> (abs_u, abs_v)
  end

(********************************************************************)
(* A demonstration program. *)

implement
main0 () =
  begin
    (* Unsigned integers. *)
    assertloc (gcd (0U, 10U) = 10U);
    assertloc (gcd (9UL, 6UL) = 3UL);
    assertloc (gcd (40902ULL, 24140ULL) = 34ULL);

    (* Signed integers. *)
    assertloc (gcd (0, 10) = gcd (0U, 10U));
    assertloc (gcd (~10, 0) = gcd (0U, 10U));
    assertloc (gcd (~6L, ~9L) = 3UL);
    assertloc (gcd (40902LL, 24140LL) = 34ULL);
    assertloc (gcd (40902LL, ~24140LL) = 34ULL);
    assertloc (gcd (~40902LL, 24140LL) = 34ULL);
    assertloc (gcd (~40902LL, ~24140LL) = 34ULL);
    assertloc (gcd (24140LL, 40902LL) = 34ULL);
    assertloc (gcd (~24140LL, 40902LL) = 34ULL);
    assertloc (gcd (24140LL, ~40902LL) = 34ULL);
    assertloc (gcd (~24140LL, ~40902LL) = 34ULL)
  end

(********************************************************************)


Stein’s algorithm, with proof of termination[edit]

Here is an implementation of Stein’s algorithm, this time with a proof of termination. Notice that the proof is rather ‘informal’; this is practical systems programming, not an exercise in mathematical logic.

(********************************************************************)
(*

  GCD of two integers, by Stein’s algorithm:
  https://en.wikipedia.org/w/index.php?title=Binary_GCD_algorithm&oldid=1072393147

  This is an implementation with proof of termination.

  The implementations shown here require the GCC builtin functions
  for ‘count trailing zeros’. If your C compiler is GCC or another
  that supports those functions, you are fine. Otherwise, one could
  easily substitute other C code.

  Compile with ‘patscc -o gcd gcd.dats’.

*)

#define ATS_EXTERN_PREFIX "rosettacode_gcd_"
#define ATS_DYNLOADFLAG 0       (* No initialization is needed. *)

#include "share/atspre_define.hats"
#include "share/atspre_staload.hats"

(********************************************************************)
(*                                                                  *)
(* Declarations of the functions.                                   *)
(*                                                                  *)

(* g1uint_gcd_stein will be the generic template function for
   unsigned integers. *)
extern fun {tk : tkind}
g0uint_gcd_stein :
  (g0uint tk, g0uint tk) -<> g0uint tk

(* g0int_gcd_stein will be the generic template function for
   signed integers, giving an unsigned result. *)
extern fun {tk_signed, tk_unsigned : tkind}
g0int_gcd_stein :
  (g0int tk_signed, g0int tk_signed) -<> g0uint tk_unsigned

(* Let us call these functions ‘gcd_stein’ or just ‘gcd’. *)
overload gcd_stein with g0uint_gcd_stein
overload gcd_stein with g0int_gcd_stein
overload gcd with gcd_stein

(********************************************************************)
(*                                                                  *)
(* The implementations.                                             *)
(*                                                                  *)

%{^

/*

  We will need a ‘count trailing zeros of a positive number’ function,
  but this is not provided in the ATS prelude. Here are
  implementations using GCC builtin functions. For fast alternatives
  in standard C, see
  https://www.chessprogramming.org/index.php?title=BitScan&oldid=22495#Trailing_Zero_Count

*/

ATSinline() atstype_uint
rosettacode_gcd_g0uint_ctz_uint (atstype_uint x)
{
  return __builtin_ctz (x);
}

ATSinline() atstype_ulint
rosettacode_gcd_g0uint_ctz_ulint (atstype_ulint x)
{
  return __builtin_ctzl (x);
}

ATSinline() atstype_ullint
rosettacode_gcd_g0uint_ctz_ullint (atstype_ullint x)
{
  return __builtin_ctzll (x);
}

%}

extern fun g0uint_ctz_uint : uint -<> int = "mac#%"
extern fun g0uint_ctz_ulint : ulint -<> int = "mac#%"
extern fun g0uint_ctz_ullint : ullint -<> int = "mac#%"

(* A generic template function for ‘count trailing zeros’ of
   non-dependent unsigned integers. *)
extern fun {tk : tkind} g0uint_ctz : g0uint tk -<> int

(* Link the implementations to the template function. *)
implement g0uint_ctz<uint_kind> (x) = g0uint_ctz_uint x
implement g0uint_ctz<ulint_kind> (x) = g0uint_ctz_ulint x
implement g0uint_ctz<ullint_kind> (x) = g0uint_ctz_ullint x

(* Let one call the function simply ‘ctz’. *)
overload ctz with g0uint_ctz

(* Now the actual implementation of g0uint_gcd_stein, the template
   function for the gcd of two unsigned integers. *)
implement {tk}
g0uint_gcd_stein (u, v) =
  let
    (* Make ‘t’ a shorthand for the unsigned integer types. *)
    typedef t = g0uint tk
    typedef t (i : int) = g1uint (tk, i)

    (* Use this macro to fake proof that an int is non-negative. *)
    macdef nonneg (n) = $UNSAFE.cast{intGte 0} ,(n)

    (* Looping is done by tail recursion. The value of (x_odd + y)
       must decrease towards zero, to prove termination. *)
    fun {tk : tkind}
    main_loop {x_odd, y : pos} .<x_odd + y>.
              (x_odd : t (x_odd), y : t (y)) :<>
        [d : pos] t (d) =
      let

        (* Remove twos from y, giving an odd number. Note
           gcd(x_odd,y_odd) = gcd(x_odd,y).  We do not have a
           dependent-type version of the following operation, so let
           us do it with non-dependent types. *)
        val [y_odd : int] y_odd =
          g1ofg0 ((g0ofg1 y) >> nonneg (ctz (g0ofg1 y)))

        (* Assert some things we know without proof. (You could also
           use assertloc, which inserts a runtime check.) *)
        prval _ = $UNSAFE.prop_assert {0 < y_odd} ()
        prval _ = $UNSAFE.prop_assert {y_odd <= y} ()
      in
        if x_odd = y_odd then
          x_odd
        else if y_odd < x_odd then
          main_loop (y_odd, x_odd - y_odd)
        else
          main_loop (x_odd, y_odd - x_odd)
      end

    fn
    u_and_v_both_positive (u : t, v : t) :<> t =
      let
        (* n = the number of common factors of two in u and v. *)
        val n = ctz (u lor v)

        (* Remove the common twos from u and v, giving x and y. *)
        val x = (u >> nonneg n)
        val y = (v >> nonneg n)

        (* Remove twos from x, giving an odd number.
           Note gcd(x_odd,y) = gcd(x,y). *)
        val x_odd = (x >> nonneg (ctz x))

        (* To prove termination of main_loop, we have to
           convert x_odd and y to a dependent type. *)
        val [x_odd : int] x_odd = g1ofg0 x_odd
        val [y : int] y = g1ofg0 y

        (* Assert that they are positive. (One could also use
           assertloc, which inserts a runtime check.) *)
        prval _ = $UNSAFE.prop_assert {0 < x_odd} ()
        prval _ = $UNSAFE.prop_assert {0 < y} ()

        val z = main_loop (x_odd, y)

        (* Convert back to the non-dependent type. *)
        val z = g0ofg1 z
      in
        (* Put the common factors of two back in. *)
        (z << nonneg n)
      end

    (* If v < u then swap u and v. This operation does not
       affect the gcd. *)
    val u = min (u, v)
    and v = max (u, v)
  in
    if iseqz u then
      v
    else
      u_and_v_both_positive (u, v)
  end

(* The implementation of g0int_gcd_stein, the template function for
   the gcd of two signed integers, giving an unsigned result. *)
implement {signed_tk, unsigned_tk}
g0int_gcd_stein (u, v) =
  let
    val abs_u = $UNSAFE.cast{g0uint unsigned_tk} (abs u)
    val abs_v = $UNSAFE.cast{g0uint unsigned_tk} (abs v)
  in
    g0uint_gcd_stein<unsigned_tk> (abs_u, abs_v)
  end

(********************************************************************)
(* A demonstration program. *)

implement
main0 () =
  begin
    (* Unsigned integers. *)
    assertloc (gcd (0U, 10U) = 10U);
    assertloc (gcd (9UL, 6UL) = 3UL);
    assertloc (gcd (40902ULL, 24140ULL) = 34ULL);

    (* Signed integers. *)
    assertloc (gcd (0, 10) = gcd (0U, 10U));
    assertloc (gcd (~10, 0) = gcd (0U, 10U));
    assertloc (gcd (~6L, ~9L) = 3UL);
    assertloc (gcd (40902LL, 24140LL) = 34ULL);
    assertloc (gcd (40902LL, ~24140LL) = 34ULL);
    assertloc (gcd (~40902LL, 24140LL) = 34ULL);
    assertloc (gcd (~40902LL, ~24140LL) = 34ULL);
    assertloc (gcd (24140LL, 40902LL) = 34ULL);
    assertloc (gcd (~24140LL, 40902LL) = 34ULL);
    assertloc (gcd (24140LL, ~40902LL) = 34ULL);
    assertloc (gcd (~24140LL, ~40902LL) = 34ULL)
  end

(********************************************************************)

Euclid’s algorithm, with proof of termination and correctness[edit]

Here is an implementation of Euclid’s algorithm, with a proof of correctness.

(********************************************************************)
(*

  GCD of two integers, by Euclid’s algorithm; verified with proofs.

  Compile with ‘patscc -o gcd gcd.dats’.

*)

#define ATS_DYNLOADFLAG 0       (* No initialization is needed. *)

#include "share/atspre_define.hats"
#include "share/atspre_staload.hats"

(********************************************************************)
(*                                                                  *)
(* Definition of the gcd by axioms in the static language.          *)
(*                                                                  *)
(* (‘Props’ are better supported in ATS, but I enjoy using the      *)
(* the static language in proofs.)                                  *)
(*                                                                  *)

(* Write the gcd as an undefined static function. It will be defined
   implicitly by axioms. (Such a function also can be used with an
   external SMT solver such as CVC4, but using an external solver is
   not the topic of this program.) *)
stacst gcd (u : int, v : int) : int

(*
   I think the reader will accept the following axioms as valid,
   if gcd(0, 0) is to be defined as equal to zero.

   (An exercise for the reader is to prove ‘gcd_of_remainder’
   from gcd (u, v) == gcd (u, v - u). This requires definitions
   of multiplication and Euclidean division, which are encoded
   in terms of props in ‘prelude/SATS/arith_prf.sats’.)
*)

extern praxi
gcd_of_zero :
  {u, v : int | u == 0; 0 <= v} (* For all integers u = 0,
                                   v non-negative. *)
  () -<prf> [gcd (u, v) == v] void

extern praxi
gcd_of_remainder :
  {u, v : int | 0 < u; 0 <= v}  (* For all integers u positive,
                                   v non-negative. *)
  () -<prf> [gcd (u, v) == gcd (u, v mod u)] void

extern praxi
gcd_is_commutative :
  {u, v : int}                  (* For all integers u, v. *)
  () -<prf> [gcd (u, v) == gcd (v, u)] void

extern praxi
gcd_of_the_absolute_values :
  {u, v : int}                  (* For all integers u, v. *)
  () -<prf> [gcd (u, v) == gcd (abs u, abs v)] void

extern praxi
gcd_is_a_function :
  {u1, v1 : int}
  {u2, v2 : int | u1 == u2; v1 == v2}
  () -<prf> [gcd (u1, v1) == gcd (u2, v2)] void

(********************************************************************)
(*                                                                  *)
(* Function declarations.                                           *)
(*                                                                  *)

(* g1uint_gcd_euclid will be the generic template function for
   unsigned integers. *)
extern fun {tk : tkind}
g1uint_gcd_euclid :
  {u, v : int}
  (g1uint (tk, u),
   g1uint (tk, v)) -<>
    g1uint (tk, gcd (u, v))

(* g1int_gcd_euclid will be the generic template function for
   signed integers, giving an unsigned result. *)
extern fun {tk_signed, tk_unsigned : tkind}
g1int_gcd_euclid :
  {u, v : int}
  (g1int (tk_signed, u),
   g1int (tk_signed, v)) -<>
    g1uint (tk_unsigned, gcd (u, v))

(* Let us call these functions ‘gcd_euclid’ or just ‘gcd’. *)
overload gcd_euclid with g1uint_gcd_euclid
overload gcd_euclid with g1int_gcd_euclid
overload gcd with gcd_euclid

(********************************************************************)
(*                                                                  *)
(* Function implementations.                                        *)
(*                                                                  *)

(* The implementation of the remainder function in the ATS2 prelude
   is inconvenient for us; it does not say that the result equals
   ‘u mod v’. Let us reimplement it more to our liking. *)
fn {tk : tkind}
g1uint_rem {u, v : int | v != 0}
           (u    : g1uint (tk, u),
            v    : g1uint (tk, v)) :<>
    [w : int | 0 <= w; w < v; w == u mod v]
    g1uint (tk, w) =
  let
    prval _ = lemma_g1uint_param u
    prval _ = lemma_g1uint_param v
  in
    $UNSAFE.cast (g1uint_mod (u, v))
  end

implement {tk}
g1uint_gcd_euclid {u, v} (u, v) =
  let
    (* The static variable v, which is defined within the curly
       braces, must, with each iteration, approach zero without
       passing it. Otherwise the loop is not proven to terminate,
       and the typechecker will reject it. *)
    fun
    loop {u, v : int | 0 <= u; 0 <= v} .<v>.
         (u    : g1uint (tk, u),
          v    : g1uint (tk, v)) :<>
        g1uint (tk, gcd (u, v)) =
      if v = g1i2u 0 then
        let
          (* prop_verify tests whether what we believe we have
             proven has actually been proven. Using it a lot lengthens
             the code but is excellent documentation. *)
          prval _ = prop_verify {0 <= u} ()
          prval _ = prop_verify {v == 0} ()

          prval _ = gcd_of_zero {v, u} ()
          prval _ = prop_verify {gcd (v, u) == u} ()

          prval _ = gcd_is_commutative {u, v} ()
          prval _ = prop_verify {gcd (u, v) == gcd (v, u)} ()

          (* Therefore, by transitivity of equality: *)
          prval _ = prop_verify {gcd (u, v) == u} ()
        in
          u
        end
      else
        let
          prval _ = prop_verify {0 <= u} ()
          prval _ = prop_verify {0 < v} ()

          prval _ = gcd_of_remainder {v, u} ()
          prval _ = prop_verify {gcd (v, u) == gcd (v, u mod v)} ()

          prval _ = gcd_is_commutative {u, v} ()
          prval _ = prop_verify {gcd (u, v) == gcd (v, u)} ()

          (* Therefore, by transitivity of equality: *)
          prval _ = prop_verify {gcd (u, v) == gcd (v, u mod v)} ()

          val [w : int] w = g1uint_rem (u, v)
          prval _ = prop_verify {0 <= w} ()
          prval _ = prop_verify {w < v} ()
          prval _ = prop_verify {w == u mod v} ()

          (* It has been proven that the function will terminate: *)
          prval _ = prop_verify {0 <= w && w < v} ()

          prval _ = gcd_is_a_function {v, u mod v} {v, w} ()
          prval _ = prop_verify {gcd (v, u mod v) == gcd (v, w)} ()

          (* Therefore, by transitivity of equality: *)
          prval _ = prop_verify {gcd (u, v) == gcd (v, w)} ()
        in
          loop (v, w)
        end

    (* u is unsigned, thus proving 0 <= u. *)
    prval _ = lemma_g1uint_param (u)

    (* v is unsigned, thus proving 0 <= v. *)
    prval _ = lemma_g1uint_param (v)
  in
    loop (u, v)
  end

implement {tk_signed, tk_unsigned}
g1int_gcd_euclid {u, v} (u, v) =
  let
    (* Prove that gcd(abs u, abs v) equals gcd(u, v). *)
    prval _ = gcd_of_the_absolute_values {u, v} ()
  in
    (* Compute gcd(abs u, abs v). The ‘g1i2u’ notations cast the
       values from signed integers to unsigned integers. *)
    g1uint_gcd_euclid (g1i2u (abs u), g1i2u (abs v))
  end

(********************************************************************)
(*                                                                  *)
(* A demonstration program.                                         *)
(*                                                                  *)
(* Unfortunately, the ATS prelude may not include implementations   *)
(* of all the functions we need for long and long long integers.    *)
(* Thus the demonstration will be entirely on regular int and uint. *)
(*                                                                  *)
(* (Including implementations here would distract from the purpose. *)
(*                                                                  *)

implement
main0 () =
  begin
    (* Unsigned integers. *)
    assertloc (gcd (0U, 10U) = 10U);
    assertloc (gcd (9U, 6U) = 3U);
    assertloc (gcd (40902U, 24140U) = 34U);

    (* Signed integers. *)
    assertloc (gcd (0, 10) = gcd (0U, 10U));
    assertloc (gcd (~10, 0) = gcd (0U, 10U));
    assertloc (gcd (~6, ~9) = 3U);
    assertloc (gcd (40902, 24140) = 34U);
    assertloc (gcd (40902, ~24140) = 34U);
    assertloc (gcd (~40902, 24140) = 34U);
    assertloc (gcd (~40902, ~24140) = 34U);
    assertloc (gcd (24140, 40902) = 34U);
    assertloc (gcd (~24140, 40902) = 34U);
    assertloc (gcd (24140, ~40902) = 34U);
    assertloc (gcd (~24140, ~40902) = 34U)
  end

(********************************************************************)

Some proofs about the gcd[edit]

For the sake of interest, here is some use of ATS's "props"-based proof system. There is no executable code in the following.

(* Typecheck this file with ‘patscc -tcats gcd-proofs.dats’. *)

(* Definition of the gcd by Euclid’s algorithm, using subtractions;
   gcd(0,0) is defined to equal zero. (I do not prove that this
   definition is equivalent to the common meaning of ‘greatest common
   divisor’; that’s not a sort of thing ATS is good at.) *)
dataprop GCD (int, int, int) =
| GCD_0_0 (0, 0, 0)
| {u : pos}
  GCD_u_0 (u, 0, u)
| {v : pos}
  GCD_0_v (0, v, v)
| {u, v : pos | u <= v}
  {d    : pos}
  GCD_u_le_v (u, v, d) of
    GCD (u, v - u, d)
| {u, v : pos | u > v}
  {d    : pos}
  GCD_u_gt_v (u, v, d) of
    GCD (u - v, v, d)
| {u, v : int | u < 0 || v < 0}
  {d : pos}
  GCD_u_or_v_neg (u, v, d) of
    GCD (abs u, abs v, d)

(* Here is a proof, by construction, of the proposition
   ‘The gcd of 12 and 8 is 4’. *)
prfn
gcd_12_8 () :<prf>
    GCD (12, 8, 4)  =
  let
    prval pf = GCD_u_0 {4} ()
    prval pf = GCD_u_le_v {4, 4} {4} (pf)
    prval pf = GCD_u_le_v {4, 8} {4} (pf)
    prval pf = GCD_u_gt_v {12, 8} {4} (pf)
  in
    pf
  end

(* A lemma: the gcd is total. That is, it is defined for all
   integers. *)
extern prfun
gcd_istot :
  {u, v : int}
  () -<prf>
    [d : int]
    GCD (u, v, d)

(* Another lemma: the gcd is a function: it has a unique value for
   any given pair of arguments. *)
extern prfun
gcd_isfun :
  {u, v : int}
  {d, e : int}
  (GCD (u, v, d),
   GCD (u, v, e)) -<prf>
    [d == e] void

(* Proof of gcd_istot. This source file will not pass typechecking
   unless the proof is valid. *)
primplement
gcd_istot {u, v} () =
  let
    prfun
    gcd_istot__nat_nat__ {u, v : nat | u != 0 || v != 0} .<u + v>.
                         () :<prf> [d : pos] GCD (u, v, d) =
      sif v == 0 then
        GCD_u_0 ()
      else sif u == 0 then
        GCD_0_v ()
      else sif u <= v then
        GCD_u_le_v (gcd_istot__nat_nat__ {u, v - u} ())
      else
        GCD_u_gt_v (gcd_istot__nat_nat__ {u - v, v} ())

    prfun
    gcd_istot__int_int__ {u, v : int | u != 0 || v != 0} .<>.
                         () :<prf> [d : pos] GCD (u, v, d) =
      sif u < 0 || v < 0 then
        GCD_u_or_v_neg (gcd_istot__nat_nat__ {abs u, abs v} ())
      else
        gcd_istot__nat_nat__ {u, v} ()
  in
    sif u == 0 && v == 0 then
      GCD_0_0 ()
    else
      gcd_istot__int_int__ {u, v} ()
  end

(* Proof of gcd_isfun. This source file will not pass typechecking
   unless the proof is valid. *)
primplement
gcd_isfun {u, v} {d, e} (pfd, pfe) =
  let
    prfun
    gcd_isfun__nat_nat__ {u, v : nat}
                         {d, e : int}
                         .<u + v>.
                         (pfd  : GCD (u, v, d),
                          pfe  : GCD (u, v, e)) :<prf> [d == e] void =
      case+ pfd of
      | GCD_0_0 () =>
        {
          prval GCD_0_0 () = pfe
        }
      | GCD_u_0 () =>
        {
          prval GCD_u_0 () = pfe
        }
      | GCD_0_v () =>
        {
          prval GCD_0_v () = pfe
        }
      | GCD_u_le_v pfd1 =>
        {
          prval GCD_u_le_v pfe1 = pfe
          prval _ = gcd_isfun__nat_nat__ (pfd1, pfe1)
        }
      | GCD_u_gt_v pfd1 =>
        {
          prval GCD_u_gt_v pfe1 = pfe
          prval _ = gcd_isfun__nat_nat__ (pfd1, pfe1)
        }
  in
    sif u < 0 || v < 0 then
      {
        prval GCD_u_or_v_neg pfd1 = pfd
        prval GCD_u_or_v_neg pfe1 = pfe
        prval _ = gcd_isfun__nat_nat__ (pfd1, pfe1)
      }
    else
      gcd_isfun__nat_nat__ (pfd, pfe)
  end

AutoHotkey[edit]

Contributed by Laszlo on the ahk forum

GCD(a,b) {
   Return b=0 ? Abs(a) : Gcd(b,mod(a,b))
}

Significantly faster than recursion:

GCD(a, b) {
    while b
        b := Mod(a | 0x0, a := b)
    return a
}

AutoIt[edit]

_GCD(18, 12)
_GCD(1071, 1029)
_GCD(3528, 3780)

Func _GCD($ia, $ib)
	Local $ret = "GCD of " & $ia & " : " & $ib & " = "
	Local $imod
	While True
		$imod = Mod($ia, $ib)
		If $imod = 0 Then Return ConsoleWrite($ret & $ib & @CRLF)
		$ia = $ib
		$ib = $imod
	WEnd
EndFunc   ;==>_GCD
Output:
GCD of 18 : 12 = 6
GCD of 1071 : 1029 = 21
GCD of 3528 : 3780 = 252

AWK[edit]

The following scriptlet defines the gcd() function, then reads pairs of numbers from stdin, and reports their gcd on stdout.

$ awk 'function gcd(p,q){return(q?gcd(q,(p%q)):p)}{print gcd($1,$2)}'
12 16
4
22 33
11
45 67
1

Axe[edit]

Lbl GCD
r₁→A
r₂→B
!If B
 A
 Return
End
GCD(B,A^B)

BASIC[edit]

Applesoft BASIC[edit]

0 A = ABS(INT(A))
1 B = ABS(INT(B))
2 GCD = A * NOT NOT B
3 FOR B = B + A * NOT B TO 0 STEP 0
4     A = GCD
5     GCD = B
6     B = A - INT (A / GCD) * GCD
7 NEXT B

BASIC256[edit]

Translation of: FreeBASIC

Iterative[edit]

function gcdI(x, y)
	while y
		t = y
		y = x mod y
		x = t
	end while

	return x
end function

# ------ test ------
a = 111111111111111
b = 11111

print : print "GCD(";a;", ";b;") = "; gcdI(a, b)
print : print "GCD(";a;", 111) = "; gcdI(a, 111)
end
Output:
Igual que la entrada de FreeBASIC.

Recursive[edit]

function gcdp(a, b)
	if b = 0 then return a
	return gcdp(b, a mod b)
end function

function gcdR(a, b)
	return gcdp(abs(a), abs(b))
end function

BBC BASIC[edit]

      DEF FN_GCD_Iterative_Euclid(A%, B%)
      LOCAL C%
      WHILE B%
        C% = A%
        A% = B%
        B% = C% MOD B%
      ENDWHILE
      = ABS(A%)

FreeBASIC[edit]

Iterative solution[edit]

' version 17-06-2015
' compile with: fbc -s console

Function gcd(x As ULongInt, y As ULongInt) As ULongInt
    Dim As ULongInt t
    While y
        t = y
        y = x Mod y
        x = t
    Wend
    Return x
End Function

' ------=< MAIN >=------

Dim As ULongInt a = 111111111111111
Dim As ULongInt b = 11111

Print : Print "GCD(";a;", ";b;") = "; gcd(a, b)
Print : Print "GCD(";a;", 111) = "; gcd(a, 111)

' empty keyboard buffer
While InKey <> "" : Wend
Print : Print : Print "hit any key to end program"
Sleep
End
Output:
GCD(111111111111111, 11111) = 11111
GCD(111111111111111, 111) = 111

Recursive solution[edit]

function gcdp( a as uinteger, b as uinteger ) as uinteger
    if b = 0 then return a
    return gcdp( b, a mod b )
end function

function gcd(a as integer, b as integer) as uinteger
    return gcdp( abs(a), abs(b) )
end function

FutureBasic[edit]

window 1, @"Greatest Common Divisor", (0,0,480,270)

local fn gcd( a as short, b as short ) as short
  short result
  
  if ( b != 0 )
    result = fn gcd( b, a mod b)
  else
    result = abs(a)
  end if
end fn = result

print fn gcd( 6, 9 )

HandleEvents

GFA Basic[edit]

'
' Greatest Common Divisor
'
a%=24
b%=112
PRINT "GCD of ";a%;" and ";b%;" is ";@gcd(a%,b%)
'
' Function computes gcd
'
FUNCTION gcd(a%,b%)
  LOCAL t%
  '
  WHILE b%<>0
    t%=a%
    a%=b%
    b%=t% MOD b%
  WEND
  '
  RETURN ABS(a%)
ENDFUNC

GW-BASIC[edit]

10 INPUT A, B
20 IF A < 0 THEN A = -A
30 IF B < 0 THEN B = -B
40 GOTO 70
50 PRINT A
60 END
70 IF B = 0 THEN GOTO 50
80 TEMP = B
90 B = A MOD TEMP
100 A = TEMP
110 GOTO 70

IS-BASIC[edit]

100 DEF GCD(A,B)
110   DO WHILE B>0
120     LET T=B
130     LET B=MOD(A,B)
140     LET A=T
150   LOOP 
160   LET GCD=A
170 END DEF 
180 PRINT GCD(12,16)

Liberty BASIC[edit]

Works with: Just BASIC
'iterative Euclid algorithm
print GCD(-2,16)
end

function GCD(a,b)
    while b
        c = a
        a = b
        b = c mod b
    wend
    GCD = abs(a)
end function

PureBasic[edit]

Iterative[edit]

Procedure GCD(x, y)
  Protected r
  While y <> 0
    r = x % y
    x = y
    y = r
  Wend
  ProcedureReturn y
EndProcedure

Recursive[edit]

Procedure GCD(x, y)
  Protected r
  r = x % y
  If (r > 0)
    y = GCD(y, r)
  EndIf
  ProcedureReturn y
EndProcedure

QuickBASIC[edit]

Works with: QuickBASIC version 4.5

Iterative[edit]

DECLARE FUNCTION gcd (a%, b%)
PRINT gcd(18, 30)
END

FUNCTION gcd (a%, b%)
  WHILE b% <> 0
    t% = b%
    b% = a% MOD b%
    a% = t%
  WEND
  gcd = ABS(a%)
END FUNCTION
Output:
 6

Recursive[edit]

DECLARE FUNCTION gcd (a%, b%)
PRINT gcd(30, 18)
END

FUNCTION gcd (a%, b%)
  IF b% = 0 THEN
    gcd = ABS(a%)
  ELSE
    gcd = gcd(b%, a% MOD b%)
  END IF
END FUNCTION
Output:
 6

Run BASIC[edit]

Works with: Just BASIC
print abs(gcd(-220,160))
function gcd(gcd,b)
    while b
        c   = gcd
        gcd = b
        b   = c mod b
    wend
end function

S-BASIC[edit]

rem - return p mod q
function mod(p, q = integer) = integer
end = p - q * (p / q)

rem - return greatest common divisor of x and y
function gcd(x, y = integer) = integer
  var r, temp = integer
  if x < y then
    begin
      temp = x
      x = y
      y = temp
    end
  r = mod(x, y)
  while r <> 0 do
    begin
      x = y
      y = r
      r = mod(x, y)
    end
end =  y

rem - exercise the function

print "The GCD of 21 and 35 is"; gcd(21,35)
print "The GCD of 23 and 35 is"; gcd(23,35)
print "The GCD of 1071 and 1029 is"; gcd(1071, 1029)
print "The GCD of 3528 and 3780 is"; gcd(3528,3780)

end
Output:
The GCD of 21 and 35 is 7
The GCD of 23 and 35 is 1
The GCD of 1071 and 1029 is 21
The GCD of 3528 and 3780 is 252

Sinclair ZX81 BASIC[edit]

 10 LET M=119
 20 LET N=544
 30 LET R=M-N*INT (M/N)
 40 IF R=0 THEN GOTO 80
 50 LET M=N
 60 LET N=R
 70 GOTO 30
 80 PRINT N
Output:
17

TI-83 BASIC, TI-89 BASIC[edit]

gcd(A,B)

The ) can be omitted in TI-83 basic

Tiny BASIC[edit]

Works with: TinyBasic
10 PRINT "First number"
20 INPUT A
30 PRINT "Second number"
40 INPUT B
50 IF A<0 THEN LET A=-A
60 IF B<0 THEN LET B=-B
70 IF A>B THEN GOTO 130
80 LET B = B - A
90 IF A=0 THEN GOTO 110
100 GOTO 50
110 PRINT B
120 END
130 LET C=A
140 LET A=B
150 LET B=C
160 GOTO 70

True BASIC[edit]

Translation of: FreeBASIC
REM Iterative solution
FUNCTION gcdI(x, y)
    DO WHILE y > 0
       LET t = y
       LET y = remainder(x, y)
       LET x = t
    LOOP
    LET gcdI = x
END FUNCTION

LET a = 111111111111111
LET b = 11111
PRINT
PRINT "GCD(";a;", ";b;") = "; gcdI(a, b)
PRINT
PRINT "GCD(";a;", 111) = "; gcdI(a, 111)
END

uBasic/4tH[edit]

Translation of: BBC BASIC
Print "GCD of 18 : 12 = "; FUNC(_GCD_Iterative_Euclid(18,12))
Print "GCD of 1071 : 1029 = "; FUNC(_GCD_Iterative_Euclid(1071,1029))
Print "GCD of 3528 : 3780 = "; FUNC(_GCD_Iterative_Euclid(3528,3780))

End

_GCD_Iterative_Euclid Param(2)
  Local (1)
  Do While b@
    c@ = a@
    a@ = b@
    b@ = c@ % b@
  Loop
Return (Abs(a@))
Output:
GCD of 18 : 12 = 6
GCD of 1071 : 1029 = 21
GCD of 3528 : 3780 = 252

0 OK, 0:205

VBA[edit]

Function gcd(u As Long, v As Long) As Long
    Dim t As Long
    Do While v
        t = u
        u = v
        v = t Mod v
    Loop
    gcd = u
End Function

This function uses repeated subtractions. Simple but not very efficient.

Public Function GCD(a As Long, b As Long) As Long
While a <> b
  If a > b Then a = a - b Else b = b - a
Wend
GCD = a
End Function
Output:

Example:

print GCD(1280, 240)
 80 
print GCD(3475689, 23566319)
 7
a=123456789
b=234736437
print GCD((a),(b))
 3 

A note on the last example: using brackets forces a and b to be evaluated before GCD is called. Not doing this will cause a compile error because a and b are not the same type as in the function declaration (they are Variant, not Long). Alternatively you can use the conversion function CLng as in print GCD(CLng(a),CLng(b))

VBScript[edit]

Function GCD(a,b)
	Do
		If a Mod b > 0 Then
			c = a Mod b
			a = b
			b = c
		Else
			GCD = b
			Exit Do
		End If
	Loop
End Function

WScript.Echo "The GCD of 48 and 18 is " & GCD(48,18) & "."
WScript.Echo "The GCD of 1280 and 240 is " & GCD(1280,240) & "."
WScript.Echo "The GCD of 1280 and 240 is " & GCD(3475689,23566319) & "."
WScript.Echo "The GCD of 1280 and 240 is " & GCD(123456789,234736437) & "."
Output:
The GCD of 48 and 18 is 6.
The GCD of 1280 and 240 is 80.
The GCD of 1280 and 240 is 7.
The GCD of 1280 and 240 is 3.

Visual Basic[edit]

Works with: Visual Basic version 5
Works with: Visual Basic version 6
Works with: VBA version 6.5
Works with: VBA version 7.1
Function GCD(ByVal a As Long, ByVal b As Long) As Long
Dim h As Long

    If a Then
        If b Then
            Do
                h = a Mod b
                a = b
                b = h
            Loop While b
        End If
        GCD = Abs(a)
    Else
        GCD = Abs(b)
    End If
    
End Function

Sub Main()
' testing the above function

  Debug.Assert GCD(12, 18) = 6
  Debug.Assert GCD(1280, 240) = 80
  Debug.Assert GCD(240, 1280) = 80
  Debug.Assert GCD(-240, 1280) = 80
  Debug.Assert GCD(240, -1280) = 80
  Debug.Assert GCD(0, 0) = 0
  Debug.Assert GCD(0, 1) = 1
  Debug.Assert GCD(1, 0) = 1
  Debug.Assert GCD(3475689, 23566319) = 7
  Debug.Assert GCD(123456789, 234736437) = 3
  Debug.Assert GCD(3780, 3528) = 252
  
End Sub

XBasic[edit]

Works with: Windows XBasic
' Greatest common divisor
PROGRAM "gcddemo"
VERSION "0.001"

DECLARE FUNCTION Entry()
DECLARE FUNCTION GcdRecursive(u&, v&)
DECLARE FUNCTION GcdIterative(u&, v&)
DECLARE FUNCTION GcdBinary(u&, v&)

FUNCTION Entry()
  m& = 49865
  n& = 69811
  PRINT "GCD("; LTRIM$(STR$(m&)); ","; n&; "):"; GcdIterative(m&, n&); " (iterative)"
  PRINT "GCD("; LTRIM$(STR$(m&)); ","; n&; "):"; GcdRecursive(m&, n&); " (recursive)"
  PRINT "GCD("; LTRIM$(STR$(m&)); ","; n&; "):"; GcdBinary (m&, n&); " (binary)"
END FUNCTION

FUNCTION GcdRecursive(u&, v&)
  IF u& MOD v& <> 0 THEN
    RETURN GcdRecursive(v&, u& MOD v&)
  ELSE
    RETURN v&
  END IF
END FUNCTION

FUNCTION GcdIterative(u&, v&)
  DO WHILE v& <> 0
    t& = u&
    u& = v&
    v& = t& MOD v&
  LOOP
  RETURN ABS(u&)
END FUNCTION

FUNCTION GcdBinary(u&, v&)
  u& = ABS(u&)
  v& = ABS(v&)
  IF u& < v& THEN
    t& = u&
    u& = v&
    v& = t&
  END IF
  IF v& = 0 THEN
    RETURN u&
  ELSE
    k& = 1
    DO WHILE (u& MOD 2 = 0) && (v& MOD 2 = 0)
      u& = u& >> 1
      v& = v& >> 1
      k& = k& << 1
    LOOP
    IF u& MOD 2 = 0 THEN
      t& = u&
    ELSE
      t& = -v&
    END IF
    DO WHILE t& <> 0
      DO WHILE t& MOD 2 = 0
        t& = t& \ 2
      LOOP
      IF t& > 0 THEN
        u& = t&
      ELSE
        v& = -t&
      END IF
      t& = u& - v&
    LOOP
    RETURN u& * k&
  END IF
END FUNCTION

END PROGRAM
Output:
GCD(49865, 69811): 9973 (iterative)
GCD(49865, 69811): 9973 (recursive)
GCD(49865, 69811): 9973 (binary)

Yabasic[edit]

sub gcd(u, v)
    local t
	
    u = int(abs(u))
    v = int(abs(v))
    while(v)
        t = u
        u = v
        v = mod(t, v)
    wend
    return u
end sub

print "Greatest common divisor: ", gcd(12345, 9876)

ZX Spectrum Basic[edit]

10 FOR n=1 TO 3
20 READ a,b
30 PRINT "GCD of ";a;" and ";b;" = ";
40 GO SUB 70
50 NEXT n
60 STOP 
70 IF b=0 THEN PRINT ABS (a): RETURN 
80 LET c=a: LET a=b: LET b=FN m(c,b): GO TO 70
90 DEF FN m(a,b)=a-INT (a/b)*b
100 DATA 12,16,22,33,45,67

Batch File[edit]

Recursive method

:: gcd.cmd
@echo off
:gcd
if "%2" equ "" goto :instructions
if "%1" equ "" goto :instructions

if %2 equ 0 (
	set final=%1
	goto :done
)
set /a res = %1 %% %2
call :gcd %2 %res%
goto :eof

:done
echo gcd=%final%
goto :eof

:instructions
echo Syntax:
echo 	GCD {a} {b}
echo.

Bc[edit]

Works with: GNU bc
Translation of: C

Utility functions:

define even(a)
{
  if ( a % 2 == 0 ) {
    return(1);
  } else {
    return(0);
  }
}

define abs(a)
{ 
  if (a<0) {
    return(-a);
  } else {
    return(a);
  }
}

Iterative (Euclid)

define gcd_iter(u, v)
{
  while(v) {
    t = u;
    u = v;
    v = t % v;
  }
  return(abs(u));
}

Recursive

define gcd(u, v)
{
  if (v) {
    return ( gcd(v, u%v) );
  } else {
    return (abs(u));
  }
}

Iterative (Binary)

define gcd_bin(u, v)
{
  u = abs(u);
  v = abs(v);
  if ( u < v ) {
    t = u; u = v; v = t;
  }
  if ( v == 0 ) { return(u); }
  k = 1;
  while (even(u) && even(v)) {
    u = u / 2; v = v / 2;
    k = k * 2;
  }
  if ( even(u) ) {
    t = -v;
  } else {
    t = u;
  }
  while (t) {
    while (even(t)) { 
      t = t / 2;
    }
 
    if (t > 0) {
      u = t;
    } else {
      v = -t;
    }
    t = u - v;
  }
  return (u * k);    
}

BCPL[edit]

get "libhdr"

let gcd(m,n) = n=0 -> m, gcd(n, m rem n)

let show(m,n) be 
    writef("gcd(%N, %N) = %N*N", m, n, gcd(m, n))
    
let start() be
$(  show(18,12)
    show(1071,1029)
    show(3528,3780)
$)
Output:
gcd(18, 12) = 6
gcd(1071, 1029) = 21
gcd(3528, 3780) = 252

Befunge[edit]

#v&<     @.$<
:<\g05%p05:_^#

BQN[edit]

Gcd ← {𝕨(|𝕊⍟(>⟜0)⊣)𝕩}

Example:

21 Gcd 35
7

Bracmat[edit]

Bracmat uses the Euclidean algorithm to simplify fractions. The den function extracts the denominator from a fraction.

(gcd=a b.!arg:(?a.?b)&!b*den$(!a*!b^-1)^-1);

Example:

{?} gcd$(49865.69811)
{!} 9973

C[edit]

Iterative Euclid algorithm[edit]

int
gcd_iter(int u, int v) {
  if (u < 0) u = -u;
  if (v < 0) v = -v;
  if (v) while ((u %= v) && (v %= u));
  return (u + v);
}

Recursive Euclid algorithm[edit]

int gcd(int u, int v) {
return (v != 0)?gcd(v, u%v):u;
}

Iterative binary algorithm[edit]

int gcd_bin(int u, int v) {
  int t, k;

  u = u < 0 ? -u : u; /* abs(u) */
  v = v < 0 ? -v : v; 
  if (u < v) {
    t = u;
    u = v;
    v = t;
  }
  if (v == 0)
    return u;

  k = 1;
  while ((u & 1) == 0 && (v & 1) == 0) { /* u, v - even */
    u >>= 1; v >>= 1;
    k <<= 1;
  }

  t = (u & 1) ? -v : u;
  while (t) {
    while ((t & 1) == 0) 
      t >>= 1;

    if (t > 0)
      u = t;
    else
      v = -t;

    t = u - v;
  }
  return u * k;        
}

C#[edit]

Iterative[edit]

static void Main()
{
	Console.WriteLine("GCD of {0} and {1} is {2}", 1, 1, gcd(1, 1));
	Console.WriteLine("GCD of {0} and {1} is {2}", 1, 10, gcd(1, 10));
	Console.WriteLine("GCD of {0} and {1} is {2}", 10, 100, gcd(10, 100));
	Console.WriteLine("GCD of {0} and {1} is {2}", 5, 50, gcd(5, 50));
	Console.WriteLine("GCD of {0} and {1} is {2}", 8, 24, gcd(8, 24));
	Console.WriteLine("GCD of {0} and {1} is {2}", 36, 17, gcd(36, 17));
	Console.WriteLine("GCD of {0} and {1} is {2}", 36, 18, gcd(36, 18));
	Console.WriteLine("GCD of {0} and {1} is {2}", 36, 19, gcd(36, 19));
	for (int x = 1; x < 36; x++)
	{
		Console.WriteLine("GCD of {0} and {1} is {2}", 36, x, gcd(36, x));
	}
	Console.Read();
}
 
/// <summary>
/// Greatest Common Denominator using Euclidian Algorithm
/// </summary>
static int gcd(int a, int b)
{
    while (b != 0) b = a % (a = b);
    return a;
}

Example output:

GCD of 1 and 1 is 1
GCD of 1 and 10 is 1
GCD of 10 and 100 is 10
GCD of 5 and 50 is 5
GCD of 8 and 24 is 8
GCD of 36 and 1 is 1
GCD of 36 and 2 is 2
..
GCD of 36 and 16 is 4
GCD of 36 and 17 is 1
GCD of 36 and 18 is 18
..
..
GCD of 36 and 33 is 3
GCD of 36 and 34 is 2
GCD of 36 and 35 is 1

Recursive[edit]

static void Main(string[] args)
{
	Console.WriteLine("GCD of {0} and {1} is {2}", 1, 1, gcd(1, 1));
	Console.WriteLine("GCD of {0} and {1} is {2}", 1, 10, gcd(1, 10));
	Console.WriteLine("GCD of {0} and {1} is {2}", 10, 100, gcd(10, 100));
	Console.WriteLine("GCD of {0} and {1} is {2}", 5, 50, gcd(5, 50));
	Console.WriteLine("GCD of {0} and {1} is {2}", 8, 24, gcd(8, 24));
	Console.WriteLine("GCD of {0} and {1} is {2}", 36, 17, gcd(36, 17));
	Console.WriteLine("GCD of {0} and {1} is {2}", 36, 18, gcd(36, 18));
	Console.WriteLine("GCD of {0} and {1} is {2}", 36, 19, gcd(36, 19));
	for (int x = 1; x < 36; x++)
	{
		Console.WriteLine("GCD of {0} and {1} is {2}", 36, x, gcd(36, x));
	}
	Console.Read();
}
 
// Greatest Common Denominator using Euclidian Algorithm
// Gist: https://gist.github.com/SecretDeveloper/6c426f8993873f1a05f7
static int gcd(int a, int b)
{	
	return b==0 ? a : gcd(b, a % b);
}

Example output:

GCD of 1 and 1 is 1
GCD of 1 and 10 is 1
GCD of 10 and 100 is 10
GCD of 5 and 50 is 5
GCD of 8 and 24 is 8
GCD of 36 and 1 is 1
GCD of 36 and 2 is 2
..
GCD of 36 and 16 is 4
GCD of 36 and 17 is 1
GCD of 36 and 18 is 18
..
..
GCD of 36 and 33 is 3
GCD of 36 and 34 is 2
GCD of 36 and 35 is 1

C++[edit]

#include <iostream>
#include <numeric>

int main() {
    std::cout << "The greatest common divisor of 12 and 18 is " << std::gcd(12, 18) << " !\n";
}
Library: Boost
#include <boost/math/common_factor.hpp>
#include <iostream>

int main() {
   std::cout << "The greatest common divisor of 12 and 18 is " << boost::math::gcd(12, 18) << "!\n";
}
Output:
The greatest common divisor of 12 and 18 is 6!

Clojure[edit]

Euclid's Algorithm[edit]

(defn gcd 
  "(gcd a b) computes the greatest common divisor of a and b."
  [a b]
  (if (zero? b)
    a
    (recur b (mod a b))))

That recur call is the same as (gcd b (mod a b)), but makes use of Clojure's explicit tail call optimization.

This can be easily extended to work with variadic arguments:

(defn gcd*
  "greatest common divisor of a list of numbers"
  [& lst]
  (reduce gcd
          lst))

Stein's Algorithm (Binary GCD)[edit]

(defn stein-gcd [a b]
  (cond
    (zero? a) b
    (zero? b) a
    (and (even? a) (even? b)) (* 2 (stein-gcd (unsigned-bit-shift-right a 1) (unsigned-bit-shift-right b 1)))
    (and (even? a) (odd? b)) (recur (unsigned-bit-shift-right a 1) b)
    (and (odd? a) (even? b)) (recur a (unsigned-bit-shift-right b 1))
    (and (odd? a) (odd? b)) (recur (unsigned-bit-shift-right (Math/abs (- a b)) 1) (min a b))))

CLU[edit]

gcd = proc (a, b: int) returns (int)
    while b~=0 do
        a, b := b, a//b
    end
    return(a)
end gcd

start_up = proc()
    po: stream := stream$primary_input()
    as: array[int] := array[int]$[18, 1071, 3528]
    bs: array[int] := array[int]$[12, 1029, 3780]
    for i: int in array[int]$indexes(as) do
        stream$putl(po, "gcd(" || int$unparse(as[i]) || ", "
            || int$unparse(bs[i]) || ") = "
            || int$unparse(gcd(as[i], bs[i])))
    end
end start_up
Output:
gcd(18, 12) = 6
gcd(1071, 1029) = 21
gcd(3528, 3780) = 252

COBOL[edit]

       IDENTIFICATION DIVISION.
       PROGRAM-ID. GCD.

       DATA DIVISION.
       WORKING-STORAGE SECTION.
       01 A        PIC 9(10)   VALUE ZEROES.
       01 B        PIC 9(10)   VALUE ZEROES.
       01 TEMP     PIC 9(10)   VALUE ZEROES.

       PROCEDURE DIVISION.
       Begin.
           DISPLAY "Enter first number, max 10 digits."
           ACCEPT A
           DISPLAY "Enter second number, max 10 digits."
           ACCEPT B
           IF A < B
             MOVE B TO TEMP
             MOVE A TO B
             MOVE TEMP TO B
           END-IF

           PERFORM UNTIL B = 0
             MOVE A TO TEMP
             MOVE B TO A
             DIVIDE TEMP BY B GIVING TEMP REMAINDER B
           END-PERFORM
           DISPLAY "The gcd is " A
           STOP RUN.

Cobra[edit]

class Rosetta
	def gcd(u as number, v as number) as number
		u, v = u.abs, v.abs
		while v > 0
			u, v = v, u % v
		return u

	def main
		print "gcd of [12] and [8] is [.gcd(12, 8)]"
		print "gcd of [12] and [-8] is [.gcd(12, -8)]"
		print "gcd of [96] and [27] is [.gcd(27, 96)]"
		print "gcd of [51] and [34] is [.gcd(34, 51)]"

Output:

gcd of 12 and 8 is 4
gcd of 12 and -8 is 4
gcd of 96 and 27 is 3
gcd of 51 and 34 is 17

CoffeeScript[edit]

Simple recursion

gcd = (x, y) ->
  if y == 0 then x else gcd y, x % y

Since JS has no TCO, here's a version with no recursion

gcd = (x, y) ->
  [1..(Math.min x, y)].reduce (acc, v) ->
    if x % v == 0 and y % v == 0 then v else acc

Common Lisp[edit]

Common Lisp provides a gcd function.

CL-USER> (gcd 2345 5432)
7

Here is an implementation using the do macro. We call the function gcd* so as not to conflict with common-lisp:gcd.

(defun gcd* (a b)
  (do () ((zerop b) (abs a))
    (shiftf a b (mod a b))))

Here is a tail-recursive implementation.

(defun gcd* (a b)
  (if (zerop b)
       a 
      (gcd2 b (mod a b))))

The last implementation is based on the loop macro.

(defun gcd* (a b)
  (loop for x = a then y
        and y = b then (mod x y)
        until (zerop y)
        finally (return x)))

Component Pascal[edit]

BlackBox Component Builder

MODULE Operations;
IMPORT StdLog,Args,Strings;

PROCEDURE Gcd(a,b: LONGINT):LONGINT;
VAR
	r: LONGINT;
BEGIN
	LOOP
		r := a MOD b;
		IF r = 0 THEN RETURN b END;
		a := b;b := r
	END
END Gcd;

PROCEDURE DoGcd*;
VAR
	x,y,done: INTEGER;
	p: Args.Params;
BEGIN
	Args.Get(p);
	IF p.argc >= 2 THEN 
		Strings.StringToInt(p.args[0],x,done);
		Strings.StringToInt(p.args[1],y,done);
		StdLog.String("gcd("+p.args[0]+","+p.args[1]+")=");StdLog.Int(Gcd(x,y));StdLog.Ln
	END		
END DoGcd;

END Operations.

Execute:
^Q Operations.DoGcd 12 8 ~
^Q Operations.DoGcd 100 5 ~
^Q Operations.DoGcd 7 23 ~
^Q Operations.DoGcd 24 -112 ~
Output:

gcd(12 ,8 )= 4
gcd(100 ,5 )= 5
gcd(7 ,23 )= 1
gcd(24 ,-112 )= -8

D[edit]

import std.stdio, std.numeric;

long myGCD(in long x, in long y) pure nothrow @nogc {
    if (y == 0)
        return x;
    return myGCD(y, x % y);
}

void main() {
    gcd(15, 10).writeln; // From Phobos.
    myGCD(15, 10).writeln;
}
Output:
5
5

Dc[edit]

[dSa%Lard0<G]dsGx+

This code assumes that there are two integers on the stack.

dc -e'28 24 [dSa%Lard0<G]dsGx+ p'

Delphi[edit]

See #Pascal / Delphi / Free Pascal.

Draco[edit]

proc nonrec gcd(word m, n) word:
    word t;
    while n ~= 0 do
        t := m;
        m := n;
        n := t % n
    od;
    m
corp

proc nonrec show(word m, n) void:
    writeln("gcd(", m, ", ", n, ") = ", gcd(m, n))
corp

proc nonrec main() void:
    show(18, 12);
    show(1071, 1029);
    show(3528, 3780)
corp
Output:
gcd(18, 12) = 6
gcd(1071, 1029) = 21
gcd(3528, 3780) = 252

DWScript[edit]

PrintLn(Gcd(231, 210));

Output:

21

Dyalect[edit]

Translation of: Go
func gcd(a, b) {
    func bgcd(a, b, res) {
        if a == b {
            return res * a
        } else if a % 2 == 0 && b % 2 == 0 {
            return bgcd(a/2, b/2, 2*res)
        } else if a % 2 == 0 {
            return bgcd(a/2, b, res)
        } else if b % 2 == 0 {
            return bgcd(a, b/2, res)
        } else if a > b {
            return bgcd(a-b, b, res)
        } else {
            return bgcd(a, b-a, res)
        }
    }
    return bgcd(a, b, 1)
}
 
var testdata = [
    (a: 33, b: 77),
    (a: 49865, b: 69811)
]
 
for v in testdata {
    print("gcd(\(v.a), \(v.b)) = \(gcd(v.a, v.b))")
}
Output:
gcd(33, 77) = 11
gcd(49865, 69811) = 9973

E[edit]

Translation of: Python
def gcd(var u :int, var v :int) {
    while (v != 0) {
        def r := u %% v
        u := v
        v := r
    }
    return u.abs()
}

EasyLang[edit]

func gcd a b . res .
  while b <> 0
    h = b
    b = a mod b
    a = h
  .
  res = a
.
call gcd 120 35 r
print r

EDSAC order code[edit]

The EDSAC had no division instruction, so the GCD routine below has to include its own code for division.

 [Greatest common divisor for Rosetta Code.
  Program for EDSAC, Initial Orders 2.]

 [Library subroutine R2. Reads positive integers during input of orders,
    and is then overwritten (so doesn't take up any memory).
  Negative numbers can be input by adding 2^35.
  Each integer is followed by 'F', except the last is followed by '#TZ'.]
            T   45 K [store address in location 45
                    values are then accessed by code letter H]
            P  220 F [<------ address here]
  GKT20FVDL8FA40DUDTFI40FA40FS39FG@S2FG23FA5@T5@E4@E13Z
            T     #H  [Tell R2 the storage location defined above]

 [Integers to be read by R2. First item is count, then pairs for GCD algo.]
  4F 1066F 2019F 1815F 1914F 103785682F 167928761F 109876463F 177777648#TZ

 [----------------------------------------------------------------------
  Library subroutine P7.
  Prints long strictly positive integer at 0D.
  10 characters, right justified, padded left with spaces.
  Closed, even; 35 storage locations; working position 4D.]
            T   56 K
  GKA3FT26@H28#@NDYFLDT4DS27@TFH8@S8@T1FV4DAFG31@SFLDUFOFFFSFL4F
  T4DA1FA27@G11@XFT28#ZPFT27ZP1024FP610D@524D!FO30@SFL8FE22@

 [---------------------------------------------------------------
  Subroutine to return GCD of two non-negative 35-bit integers.
  Input:  Integers at 4D, 6D.
  Output: GCD at 4D; changes 6D.
  41 locations; working location 0D.]
            T  100 K
            G      K
            A    3 F  [plant link]
            T   39 @
            S    4 D  [test for 4D = 0]
            E   37 @  [if so, quick exit, GCD = 6D]
            T   40 @  [clear acc]
      [5]   A    4 D  [load divisor]
      [6]   T      D  [initialize shifted divisor]
            A    6 D  [load dividend]
            R      D  [shift 1 right]
            S      D  [shifted divisor > dividend/2 yet?]
            G   15 @  [yes, start subtraction]
            T   40 @  [no, clear acc]
            A      D  [shift divisor 1 more]
            L      D
            E    6 @  [loop back (always, since acc = 0)]
     [15]   T   40 @  [clear acc]
     [16]   A    6 D  [load remainder (initially = dividend)]
            S      D  [trial subtraction]
            G   20 @  [skip if can't subtract]
            T    6 D  [update remainder]
     [20]   T   40 @  [clear acc]
            A    4 D  [load original divisor]
            S      D  [is shifted divisor back to original?]
            E   29 @  [yes, jump out with acc = 0]
            T   40 @  [no, clear acc]
            A      D  [shift divisor 1 right]
            R      D
            T      D
            E   16 @  [loop back (always, since acc = 0)]
         [Here when finished dividing 6D by 4D.
          Remainder is at 6D; acc = 0.]
     [29]   S    6 D  [test for 6D = 0]
            E   39 @  [if so, exit with GCD in 4D]
            T      D  [else swap 4D and 6D]
            A    4 D
            T    6 D
            S      D
            T    4 D
            E    5 @  [loop back]
     [37]   A    6 D  [here if 4D = 0 at start; GCD is 6D]
            T    4 D  [return in 4D as GCD]
     [39]   E      F
     [40]   P      F  [junk word, to clear accumulator]
                             
 [----------------------------------------------------------------------
  Main routine]
            T  150 K
            G      K
  [Variable]
      [0]   P      F
  [Constants]
      [1]   P      D [single-word 1]
      [2]   A    2#H [order to load first number of first pair]
      [3]   P    2 F [to inc addresses by 2]
      [4]   #      F [figure shift]
      [5]   K 2048 F [letter shift]
      [6]   G      F [letters to print 'GCD']
      [7]   C      F
      [8]   D      F
      [9]   V      F [equals sign (in figures mode)]
     [10]   !      F [space]
     [11]   @      F [carriage return]
     [12]   &      F [line feed]
     [13]   K 4096 F [null char]
         [Enter here with acc = 0]
     [14]   O    4 @ [set teleprinter to figures]
            S      H [negative of number of pairs]
            T      @ [initialize counter]
            A    2 @ [initial load order]
     [18]   U   23 @ [plant order to load 1st integer]
            U   32 @
            A    3 @ [inc address by 2]
            U   28 @ [plant order to load 2nd integer]
            T   34 @
     [23]   A     #H [load 1st integer (order set up at runtime)]
            T      D [to 0D for printing]
            A   25 @ [for return from print subroutine]
            G   56 F [print 1st number]
            O   10 @ [followed by space]
     [28]   A     #H [load 2nd integer (order set up at runtime)]
            T      D [to 0D for printing]
            A   30 @ [for return from print subroutine]
            G   56 F [print 2nd number]
     [32]   A     #H [load 1st integer (order set up at runtime)]
            T    4 D [to 4D for GCD subroutine]
     [34]   A     #H [load 2nd integer (order set up at runtime)]
            T    6 D [to 6D for GCD subroutine]
     [36]   A   36 @ [for return from subroutine]
            G  100 F [call subroutine for GCD]
         [Cosmetic printing, add '  GCD = ']
            O   10 @
            O   10 @
            O    5 @
            O    6 @
            O    7 @
            O    8 @
            O    4 @
            O   10 @
            O    9 @
            O   10 @
            A    4 D [load GCD]
            T      D [to 0D for printing]
            A   50 @ [for return from print subroutine]
            G   56 F [print GCD]
            O   11 @ [followed by new line]
            O   12 @
          [On to next pair]
            A      @ [load negative count of c.f.s]
            A    1 @ [add 1]
            E   62 @ [exit if count = 0]
            T      @ [store back]
            A   23 @ [order to load first of pair]
            A    3 @ [inc address by 4 for next c.f.]
            A    3 @
            G   18 @ [loop back (always, since 'A' < 0)]
     [62]   O   13 @  [null char to flush teleprinter buffer]
            Z      F  [stop]
            E   14 Z  [define entry point]
            P      F  [acc = 0 on entry]
Output:
      1066       2019  GCD =          1
      1815       1914  GCD =         33
 103785682  167928761  GCD =       1001
 109876463  177777648  GCD =    1234567

Eiffel[edit]

Translation of: D
class
	APPLICATION

create
	make

feature -- Implementation

	gcd (x: INTEGER y: INTEGER): INTEGER
		do
			if y = 0 then
				Result := x
			else
				Result := gcd (y, x \\ y);
			end
		end

feature {NONE} -- Initialization

	make
			-- Run application.
		do
			print (gcd (15, 10))
			print ("%N")
		end

end

Elena[edit]

Translation of: C#

ELENA 4.x :

import system'math;
import extensions;
 
gcd(a,b)
{
    var i := a;
    var j := b;
    while(j != 0)
    {
        var tmp := i;
        i := j;
        j := tmp.mod(j)
    };
 
    ^ i
}
 
printGCD(a,b)
{
    console.printLineFormatted("GCD of {0} and {1} is {2}", a, b, gcd(a,b))
}
 
public program()
{
    printGCD(1,1);
    printGCD(1,10);
    printGCD(10,100);
    printGCD(5,50);
    printGCD(8,24);
    printGCD(36,17);
    printGCD(36,18);
    printGCD(36,19);
    printGCD(36,33);
}
Output:
GCD of 1 and 1 is 1
GCD of 1 and 10 is 1
GCD of 10 and 100 is 10
GCD of 5 and 50 is 5
GCD of 8 and 24 is 8
GCD of 36 and 17 is 1
GCD of 36 and 18 is 18
GCD of 36 and 19 is 1
GCD of 36 and 33 is 3

Elixir[edit]

defmodule RC do
  def gcd(a,0), do: abs(a)
  def gcd(a,b), do: gcd(b, rem(a,b))
end

IO.puts RC.gcd(1071, 1029)
IO.puts RC.gcd(3528, 3780)
Output:
21
252

Emacs Lisp[edit]

(defun gcd (a b)
  (cond
   ((< a b) (gcd a (- b a)))
   ((> a b) (gcd (- a b) b))
   (t a)))

Erlang[edit]

% Implemented by Arjun Sunel
-module(gcd).
-export([main/0]).

main() ->gcd(-36,4).
	
gcd(A, 0) -> A;

gcd(A, B) -> gcd(B, A rem B).
Output:
4

ERRE[edit]

This is a iterative version.

PROGRAM EUCLIDE
! calculate G.C.D. between two integer numbers
! using Euclidean algorithm

!VAR J%,K%,MCD%,A%,B%

BEGIN
  PRINT(CHR$(12);"Input two numbers : ";)  !CHR$(147) in C-64 version
  INPUT(J%,K%)
  A%=J% B%=K%
  WHILE A%<>B% DO
    IF A%>B%
       THEN
         A%=A%-B%
       ELSE
         B%=B%-A%
    END IF
  END WHILE
  MCD%=A%
  PRINT("G.C.D. between";J%;"and";K%;"is";MCD%)
END PROGRAM
Output:
Input two numbers : ? 112,44
G.C.D. between 112 and 44 is 4

Euler Math Toolbox[edit]

Non-recursive version in Euler Math Toolbox. Note, that there is a built-in command.

>ggt(123456795,1234567851)
 33
>function myggt (n:index, m:index) ...
$  if n<m then {n,m}={m,n}; endif;
$  repeat
$    k=mod(n,m);
$    if k==0 then return m; endif;
$    if k==1 then return 1; endif;
$    {n,m}={m,k};
$  end;
$  endfunction
>myggt(123456795,1234567851)
 33

Euphoria[edit]

Translation of: C/C++

Iterative Euclid algorithm[edit]

function gcd_iter(integer u, integer v)
    integer t
    while v do
        t = u
        u = v
        v = remainder(t, v)
    end while
    if u < 0 then
        return -u
    else
        return u
    end if
end function

Recursive Euclid algorithm[edit]

function gcd(integer u, integer v)
    if v then
        return gcd(v, remainder(u, v))
    elsif u < 0 then
        return -u
    else
        return u
    end if
end function

Iterative binary algorithm[edit]

function gcd_bin(integer u, integer v)
    integer t, k
    if u < 0 then -- abs(u)
        u = -u
    end if
    if v < 0 then -- abs(v)
        v = -v
    end if
    if u < v then
        t = u
        u = v
        v = t
    end if
    if v = 0 then
        return u
    end if
    k = 1
    while and_bits(u,1) = 0 and and_bits(v,1) = 0 do
        u = floor(u/2) -- u >>= 1
        v = floor(v/2) -- v >>= 1
        k *= 2 -- k <<= 1
    end while
    if and_bits(u,1) then
        t = -v
    else
        t = u
    end if
    while t do
        while and_bits(t, 1) = 0 do
            t = floor(t/2)
        end while
        if t > 0 then
            u = t
        else
            v = -t
        end if
        t = u - v
    end while
    return u * k
end function

Excel[edit]

Excel's GCD can handle multiple values. Type in a cell:

=GCD(A1:E1)
Sample Output:

This will get the GCD of the first 5 cells of the first row.

30	10	500	25	1000
5				

Ezhil[edit]

## இந்த நிரல் இரு எண்களுக்கு இடையிலான மீச்சிறு பொது மடங்கு (LCM), மீப்பெரு பொது வகுத்தி (GCD) என்ன என்று கணக்கிடும்

நிரல்பாகம் மீபொவ(எண்1, எண்2)

	@(எண்1 == எண்2) ஆனால்

  ## இரு எண்களும் சமம் என்பதால், அந்த எண்ணேதான் அதன் மீபொவ

		பின்கொடு எண்1

	@(எண்1 > எண்2) இல்லைஆனால்
	
		சிறியது = எண்2
		பெரியது = எண்1

	இல்லை

		சிறியது = எண்1
		பெரியது = எண்2

	முடி

	மீதம் = பெரியது % சிறியது

	@(மீதம் == 0) ஆனால்

  ## பெரிய எண்ணில் சிறிய எண் மீதமின்றி வகுபடுவதால், சிறிய எண்தான் மீப்பெரு பொதுவகுத்தியாக இருக்கமுடியும்

		பின்கொடு சிறியது

	இல்லை

		தொடக்கம் = சிறியது - 1

		நிறைவு = 1

		@(எண் = தொடக்கம், எண் >= நிறைவு, எண் = எண் - 1) ஆக

			மீதம்1 = சிறியது % எண்

			மீதம்2 = பெரியது % எண்

   ## இரு எண்களையும் மீதமின்றி வகுக்கக்கூடிய பெரிய எண்ணைக் கண்டறிகிறோம்

			@((மீதம்1 == 0) && (மீதம்2 == 0)) ஆனால்

				பின்கொடு எண்
			
			முடி

		முடி

	முடி

முடி

 = int(உள்ளீடு("ஓர் எண்ணைத் தாருங்கள் "))
 = int(உள்ளீடு("இன்னோர் எண்ணைத் தாருங்கள் "))

பதிப்பி "நீங்கள் தந்த இரு எண்களின் மீபொவ (மீப்பெரு பொது வகுத்தி, GCD) = ", மீபொவ(, )

F#[edit]

let rec gcd a b =
  if b = 0 
    then abs a
  else gcd b (a % b)
 
>gcd 400 600
val it : int = 200

Factor[edit]

: gcd ( a b -- c )
    [ abs ] [
        [ nip ] [ mod ] 2bi gcd
    ] if-zero ;

FALSE[edit]

10 15$ [0=~][$@$@$@\/*-$]#%. { gcd(10,15)=5 }

Fantom[edit]

class Main
{
  static Int gcd (Int a, Int b)
  {
    a = a.abs
    b = b.abs
    while (b > 0)
    {
      t := a
      a = b
      b = t % b
    }
    return a
  }

  public static Void main()
  {
    echo ("GCD of 51, 34 is: " + gcd(51, 34))
  }
}

Fermat[edit]

GCD(a,b)

Forth[edit]

: gcd ( a b -- n )
  begin dup while tuck mod repeat drop ;

Fortran[edit]

Works with: Fortran version 95 and later

Recursive Euclid algorithm[edit]

recursive function gcd_rec(u, v) result(gcd)
    integer             :: gcd
    integer, intent(in) :: u, v
    
    if (mod(u, v) /= 0) then
        gcd = gcd_rec(v, mod(u, v))
    else
        gcd = v
    end if
end function gcd_rec

Iterative Euclid algorithm[edit]

subroutine gcd_iter(value, u, v)
  integer, intent(out) :: value
  integer, intent(inout) :: u, v
  integer :: t

  do while( v /= 0 )
     t = u
     u = v
     v = mod(t, v)
  enddo
  value = abs(u)
end subroutine gcd_iter

A different version, and implemented as function

function gcd(v, t)
  integer :: gcd
  integer, intent(in) :: v, t
  integer :: c, b, a

  b = t
  a = v
  do
     c = mod(a, b)
     if ( c == 0) exit
     a = b
     b = c
  end do
  gcd = b ! abs(b)
end function gcd

Iterative binary algorithm[edit]

subroutine gcd_bin(value, u, v)
  integer, intent(out) :: value
  integer, intent(inout) :: u, v
  integer :: k, t

  u = abs(u)
  v = abs(v)
  if( u < v ) then
     t = u
     u = v
     v = t
  endif
  if( v == 0 ) then
     value = u
     return
  endif
  k = 1
  do while( (mod(u, 2) == 0).and.(mod(v, 2) == 0) )
     u = u / 2
     v = v / 2
     k = k * 2
  enddo
  if( (mod(u, 2) == 0) ) then
     t = u
  else
     t = -v
  endif
  do while( t /= 0 )
     do while( (mod(t, 2) == 0) )
        t = t / 2
     enddo
     if( t > 0 ) then
        u = t
     else
        v = -t
     endif
     t = u - v
  enddo
  value = u * k
end subroutine gcd_bin

Notes on performance[edit]

gcd_iter(40902, 24140) takes us about 2.8 µsec

gcd_bin(40902, 24140) takes us about 2.5 µsec

Iterative binary algorithm in Fortran 2008[edit]

Works with: Fortran version 2008
Works with: Fortran version 2018
Translation of: ATS

Fortran 2008 introduces new intrinsic functions for integer operations that nowadays usually have hardware support, such as TRAILZ to count trailing zeros.

! Stein’s algorithm implemented in Fortran 2008.
! Translated from my implementation for ATS/Postiats.

elemental function gcd (u, v) result (d)
  implicit none
  integer, intent(in) :: u, v
  integer :: d

  integer :: x, y

  ! gcd(x,y) = gcd(u,v), but x and y are non-negative and x <= y.
  x = min (abs (u), abs (v))
  y = max (abs (u), abs (v))

  if (x == 0) then
     d = y
  else
     d = gcd_pos_pos (x, y)
  end if

contains

  elemental function gcd_pos_pos (u, v) result (d)
    integer, intent(in) :: u, v
    integer :: d

    integer :: n
    integer :: x, y
    integer :: p, q

    ! n = the number of common factors of two in u and v.
    n = trailz (ior (u, v))

    ! Remove the common twos from u and v, giving x and y.
    x = ishft (u, -n)
    y = ishft (v, -n)

    ! Make both numbers odd. One of the numbers already was odd.
    ! There is no effect on the value of their gcd.
    x = ishft (x, -trailz (x))
    y = ishft (y, -trailz (y))

    do while (x /= y)
       ! If x > y then swap x and y, renaming them p
       ! and q. Thus p <= q, and gcd(p,q) = gcd(x,y).
       p = min (x, y)
       q = max (x, y)

       x = p                    ! x remains odd.
       q = q - p
       y = ishft (q, -trailz (q)) ! Make y odd again.
    end do

    ! Put the common twos back in.
    d = ishft (x, n)
  end function gcd_pos_pos

end function gcd

program test_gcd
  implicit none

  interface
     elemental function gcd (u, v) result (d)
       integer, intent(in) :: u, v
       integer :: d
     end function gcd
  end interface

  write (*, '("gcd (0, 0) = ", I0)') gcd (0, 0)
  write (*, '("gcd (0, 10) = ", I0)') gcd (0, 10)
  write (*, '("gcd (-6, -9) = ", I0)') gcd (-6, -9)
  write (*, '("gcd (64 * 5, -16 * 3) = ", I0)') gcd (64 * 5, -16 * 3)
  write (*, '("gcd (40902, 24140) = ", I0)') gcd (40902, 24140)
  write (*, '("gcd (-40902, 24140) = ", I0)') gcd (-40902, 24140)
  write (*, '("gcd (40902, -24140) = ", I0)') gcd (40902, -24140)
  write (*, '("gcd (-40902, -24140) = ", I0)') gcd (-40902, -24140)
  write (*, '("gcd (24140, 40902) = ", I0)') gcd (24140, 40902)

end program test_gcd
Output:
gcd (0, 0) = 0
gcd (0, 10) = 10
gcd (-6, -9) = 3
gcd (64 * 5, -16 * 3) = 16
gcd (40902, 24140) = 34
gcd (-40902, 24140) = 34
gcd (40902, -24140) = 34
gcd (-40902, -24140) = 34
gcd (24140, 40902) = 34

Free Pascal[edit]

See #Pascal / Delphi / Free Pascal.

Frege[edit]

module gcd.GCD where

pure native parseInt java.lang.Integer.parseInt :: String -> Int

gcd' a 0 = a
gcd' a b = gcd' b (a `mod` b)

main args = do
    (a:b:_) = args
    println $ gcd' (parseInt a) (parseInt b)

Frink[edit]

Frink has a builtin gcd[x,y] function that returns the GCD of two integers (which can be arbitrarily large.)

println[gcd[12345,98765]]

FunL[edit]

FunL has pre-defined function gcd in module integers defined as:

def
  gcd( 0, 0 ) = error( 'integers.gcd: gcd( 0, 0 ) is undefined' )
  gcd( a, b ) =
    def
      _gcd( a, 0 ) = a
      _gcd( a, b ) = _gcd( b, a%b )

    _gcd( abs(a), abs(b) )

GAP[edit]

# Built-in
GcdInt(35, 42);
# 7

# Euclidean algorithm
GcdInteger := function(a, b)
    local c;
    a := AbsInt(a);
    b := AbsInt(b);
    while b > 0 do
        c := a;
        a := b;
        b := RemInt(c, b);
    od;
    return a;
end;

GcdInteger(35, 42);
# 7

Genyris[edit]

Recursive[edit]

def gcd (u v)
    u = (abs u)
    v = (abs v)
    cond
       (equal? v 0) u
       else (gcd v (% u v))

Iterative[edit]

def gcd (u v)
    u = (abs u)
    v = (abs v)
    while (not (equal? v 0))
       var tmp (% u v)
       u = v
       v = tmp
    u

GML[edit]

 var n,m,r;
 n = max(argument0,argument1);
 m = min(argument0,argument1);
 while (m != 0) 
 {
  r = n mod m;
  n = m;
  m = r;
 }
 return a;

gnuplot[edit]

gcd (a, b) = b == 0 ? a : gcd (b, a % b)

Example:

print gcd (111111, 1111)

Output:

11

Go[edit]

Binary Euclidian[edit]

package main

import "fmt"

func gcd(a, b int) int {
    var bgcd func(a, b, res int) int

    bgcd = func(a, b, res int) int {
	switch {
	case a == b:
	    return res * a
	case a % 2 == 0 && b % 2 == 0:
	    return bgcd(a/2, b/2, 2*res)
	case a % 2 == 0:
	    return bgcd(a/2, b, res)
	case b % 2 == 0:
	    return bgcd(a, b/2, res)
	case a > b:
	    return bgcd(a-b, b, res)
	default:
	    return bgcd(a, b-a, res)
	}
    }

    return bgcd(a, b, 1)
}

func main() {
    type pair struct {
	a int
	b int
    }

    var testdata []pair = []pair{
	pair{33, 77},
	pair{49865, 69811},
    }

    for _, v := range testdata {
	fmt.Printf("gcd(%d, %d) = %d\n", v.a, v.b, gcd(v.a, v.b))
    }
}
Output for Binary Euclidian algorithm:
gcd(33, 77) = 11
gcd(49865, 69811) = 9973

Iterative[edit]

package main

import "fmt"

func gcd(x, y int) int {
    for y != 0 {
        x, y = y, x%y
    }
    return x
}

func main() {
    fmt.Println(gcd(33, 77))
    fmt.Println(gcd(49865, 69811))
}

Builtin[edit]

(This is just a wrapper for big.GCD)

package main

import (
    "fmt"
    "math/big"
)

func gcd(x, y int64) int64 {
    return new(big.Int).GCD(nil, nil, big.NewInt(x), big.NewInt(y)).Int64()
}

func main() {
    fmt.Println(gcd(33, 77))
    fmt.Println(gcd(49865, 69811))
}
Output in either case:
11
9973

Golfscript[edit]

;'2706 410'
~{.@\%.}do;
Output:
82

Groovy[edit]

Recursive[edit]

def gcdR
gcdR = { m, n -> m = m.abs(); n = n.abs(); n == 0 ? m : m%n == 0 ? n : gcdR(n, m%n) }

Iterative[edit]

def gcdI = { m, n -> m = m.abs(); n = n.abs(); n == 0 ? m : { while(m%n != 0) { t=n; n=m%n; m=t }; n }() }

Test program:

println "                R     I"
println "gcd(28, 0)   = ${gcdR(28, 0)} == ${gcdI(28, 0)}"
println "gcd(0, 28)   = ${gcdR(0, 28)} == ${gcdI(0, 28)}"
println "gcd(0, -28)  = ${gcdR(0, -28)} == ${gcdI(0, -28)}"
println "gcd(70, -28) = ${gcdR(70, -28)} == ${gcdI(70, -28)}"
println "gcd(70, 28)  = ${gcdR(70, 28)} == ${gcdI(70, 28)}"
println "gcd(28, 70)  = ${gcdR(28, 70)} == ${gcdI(28, 70)}"
println "gcd(800, 70) = ${gcdR(800, 70)} == ${gcdI(800, 70)}"
println "gcd(27, -70) =  ${gcdR(27, -70)} ==  ${gcdI(27, -70)}"

Output:

                R     I
gcd(28, 0)   = 28 == 28
gcd(0, 28)   = 28 == 28
gcd(0, -28)  = 28 == 28
gcd(70, -28) = 14 == 14
gcd(70, 28)  = 14 == 14
gcd(28, 70)  = 14 == 14
gcd(800, 70) = 10 == 10
gcd(27, -70) =  1 ==  1

Haskell[edit]

That is already available as the function gcd in the Prelude. Here's the implementation, with one name adjusted to avoid a Wiki formatting glitch:

gcd :: (Integral a) => a -> a -> a
gcd x y = gcd_ (abs x) (abs y)
  where
    gcd_ a 0 = a
    gcd_ a b = gcd_ b (a `rem` b)

HicEst[edit]

FUNCTION gcd(a, b)
   IF(b == 0) THEN
     gcd = ABS(a)
   ELSE
     aa = a
     gcd = b
     DO i = 1, 1E100
       r = ABS(MOD(aa, gcd))
       IF( r == 0 ) RETURN
       aa = gcd
       gcd = r
     ENDDO
   ENDIF
 END

Icon and Unicon[edit]

link numbers   # gcd is part of the Icon Programming Library
procedure main(args)
    write(gcd(arg[1], arg[2])) | "Usage: gcd n m")
end
numbers implements this as:
procedure gcd(i,j)		#: greatest common divisor
   local r

   if (i | j) < 1 then runerr(501)

   repeat {
      r := i % j
      if r = 0 then return j
      i := j
      j := r
      }
end

J[edit]

x+.y

For example:

   12 +. 30
6

Note that +. is a single, two character token. GCD is a primitive in J (and anyone that has studied the right kind of mathematics should instantly recognize why the same operation is used for both GCD and OR -- among other things, GCD and boolean OR both have the same identity element: 0, and of course they produce the same numeric results on the same arguments (when we are allowed to use the usual 1 bit implementation of 0 and 1 for false and true) - more than that, though, GCD corresponds to George Boole's original "Boolean Algebra" (as it was later called). The redefinition of "Boolean algebra" to include logical negation came much later, in the 20th century).

gcd could also be defined recursively, if you do not mind a little inefficiency:

gcd=: (| gcd [)^:(0<[)&|

Java[edit]

Iterative[edit]

public static long gcd(long a, long b){
   long factor= Math.min(a, b);
   for(long loop= factor;loop > 1;loop--){
      if(a % loop == 0 && b % loop == 0){
         return loop;
      }
   }
   return 1;
}

Iterative Euclid's Algorithm[edit]

public static int gcd(int a, int b) //valid for positive integers.
{
	while(b > 0)
	{
		int c = a % b;
		a = b;
		b = c;
	}
	return a;
}

Optimized Iterative[edit]

static int gcd(int a,int b)
	{
		int min=a>b?b:a,max=a+b-min, div=min;
		for(int i=1;i<min;div=min/++i)
			if(min%div==0&&max%div==0)
				return div;
		return 1;
	}

Iterative binary algorithm[edit]

Translation of: C/C++
public static long gcd(long u, long v){
  long t, k;
 
  if (v == 0) return u;
  
  u = Math.abs(u);
  v = Math.abs(v); 
  if (u < v){
    t = u;
    u = v;
    v = t;
  }
 
  for(k = 1; (u & 1) == 0 && (v & 1) == 0; k <<= 1){
    u >>= 1; v >>= 1;
  }
 
  t = (u & 1) != 0 ? -v : u;
  while (t != 0){
    while ((t & 1) == 0) t >>= 1;
 
    if (t > 0)
      u = t;
    else
      v = -t;
 
    t = u - v;
  }
  return u * k;
}

Recursive[edit]

public static long gcd(long a, long b){
   if(a == 0) return b;
   if(b == 0) return a;
   if(a > b) return gcd(b, a % b);
   return gcd(a, b % a);
}

Built-in[edit]

import java.math.BigInteger;

public static long gcd(long a, long b){
   return BigInteger.valueOf(a).gcd(BigInteger.valueOf(b)).longValue();
}

JavaScript[edit]

Iterative implementation

function gcd(a,b) {
  a = Math.abs(a);
  b = Math.abs(b);

  if (b > a) {
    var temp = a;
    a = b;
    b = temp; 
  }

  while (true) {
    a %= b;
    if (a === 0) { return b; }
    b %= a;
    if (b === 0) { return a; }
  }
}

Recursive.

function gcd_rec(a, b) {
  return b ? gcd_rec(b, a % b) : Math.abs(a);
}

Implementation that works on an array of integers.

function GCD(arr) {
  var i, y,
      n = arr.length,
      x = Math.abs(arr[0]);

  for (i = 1; i < n; i++) {
    y = Math.abs(arr[i]);

    while (x && y) {
      (x > y) ? x %= y : y %= x;
    }
    x += y;
  }
  return x;
}

//For example:
GCD([57,0,-45,-18,90,447]); //=> 3

Joy[edit]

DEFINE gcd == [0 >] [dup rollup rem] while pop.

jq[edit]

def recursive_gcd(a; b):
  if b == 0 then a 
  else recursive_gcd(b; a % b)
  end ;
Recent versions of jq include support for tail recursion optimization for arity-0 filters (which can be thought of as arity-1 functions), so here is an implementation that takes advantage of that optimization. Notice that the subfunction, rgcd, can be easily derived from recursive_gcd above by moving the arguments to the input:
def gcd(a; b):
  # The subfunction expects [a,b] as input
  # i.e. a ~ .[0] and b ~ .[1]
  def rgcd: if .[1] == 0 then .[0]
         else [.[1], .[0] % .[1]] | rgcd
         end;
  [a,b] | rgcd ;

Julia[edit]

Julia includes a built-in gcd function:

julia> gcd(4,12)
4
julia> gcd(6,12)
6
julia> gcd(7,12)
1

The actual implementation of this function in Julia 0.2's standard library is reproduced here:

function gcd{T<:Integer}(a::T, b::T)
    neg = a < 0
    while b != 0
        t = b
        b = rem(a, b)
        a = t
    end
    g = abs(a)
    neg ? -g : g
end

(For arbitrary-precision integers, Julia calls a different implementation from the GMP library.)

K[edit]

gcd:{:[~x;y;_f[y;x!y]]}

Klong[edit]

gcd::{:[~x;y:|~y;x:|x>y;.f(y;x!y);.f(x;y!x)]}

Kotlin[edit]

Recursive one line solution:

tailrec fun gcd(a: Int, b: Int): Int = if (b == 0) kotlin.math.abs(a) else gcd(b, a % b)

LabVIEW[edit]

Translation of: AutoHotkey

It may be helpful to read about Recursion in LabVIEW.
This image is a VI Snippet, an executable image of LabVIEW code. The LabVIEW version is shown on the top-right hand corner. You can download it, then drag-and-drop it onto the LabVIEW block diagram from a file browser, and it will appear as runnable, editable code.
LabVIEW Greatest common divisor.png

Lambdatalk[edit]

{def gcd
 {lambda {:a :b}
  {if {= :b 0}
   then :a
   else {gcd :b {% :a :b}}}}}
-> gcd

{gcd 12 3}
-> 3

{gcd 123 122}
-> 1

{S.map {gcd 123} {S.serie 1 30}}
-> 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3

A simpler one if a and b are greater than zero

{def GCD
 {lambda {:a :b}
  {if {= :a :b} 
   then :a
   else {if {> :a :b}
   then {GCD {- :a :b} :b}
   else {GCD :a {- :b :a}}}}}}
-> GCD

{S.map {GCD 123} {S.serie 1 30}}
-> 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3

LFE[edit]

Translation of: Clojure
> (defun gcd
  "Get the greatest common divisor."
  ((a 0) a)
  ((a b) (gcd b (rem a b))))

Usage:

> (gcd 12 8)
4
> (gcd 12 -8)
4
> (gcd 96 27)
3
> (gcd 51 34)
17

Limbo[edit]

gcd(x: int, y: int): int
{
	if(y == 0)
		return x;
	return gcd(y, x % y);
}

LiveCode[edit]

function gcd x,y
   repeat until y = 0
      put x mod y into z
      put y into x
      put z into y
   end repeat
   return x
end gcd

[edit]

to gcd :a :b
  if :b = 0 [output :a]
  output gcd :b  modulo :a :b
end

LOLCODE[edit]

HAI 1.3

HOW IZ I gcd YR a AN YR b
	a R BIGGR OF a AN PRODUKT OF a AN -1   BTW  absolute value of a
	b R BIGGR OF b AN PRODUKT OF b AN -1   BTW  absolute value of b
	BOTH SAEM a AN b, O RLY?
		YA RLY
			FOUND YR a
		OIC
	BOTH SAEM a AN 0, O RLY?
		YA RLY
			FOUND YR b
		OIC
	BOTH SAEM b AN 0, O RLY?
		YA RLY
			FOUND YR a
		OIC
	BOTH SAEM b AN BIGGR OF a AN b, O RLY?       BTW  make sure a is the larger of (a, b)
		YA RLY
			I HAS A temp ITZ a
			a R b
			b R temp
		OIC

	IM IN YR loop
		I HAS A temp ITZ b
		b R MOD OF a AN b
		a R temp
		BOTH SAEM b AN 0, O RLY?
			YA RLY
				FOUND YR a
			OIC
	IM OUTTA YR loop 
IF U SAY SO

VISIBLE I IZ gcd YR 40902 AN YR 24140 MKAY

KTHXBYE

Try it online!

LSE[edit]

(*
** MÉTHODE D'EUCLIDE POUR TROUVER LE PLUS GRAND DIVISEUR COMMUN D'UN
** NUMÉRATEUR ET D'UN DÉNOMINATEUR. 
*)
PROCÉDURE &PGDC(ENTIER U, ENTIER V) : ENTIER LOCAL U, V
    ENTIER T
    TANT QUE U > 0 FAIRE
        SI U< V ALORS
            T<-U
            U<-V
            V<-T
        FIN SI
        U <- U - V
    BOUCLER
    RÉSULTAT V
FIN PROCÉDURE

PROCÉDURE &DEMO(ENTIER U, ENTIER V) LOCAL U, V
    AFFICHER ['Le PGDC de ',U,'/',U,' est ',U,/] U, V, &PGDC(U,V)
FIN PROCÉDURE

&DEMO(9,12)
&DEMO(6144,8192)
&DEMO(100,5)
&DEMO(7,23)

Resultats:

Le PGDC de 9/12 est 3
Le PGDC de 6144/8192 est 2048
Le PGDC de 100/5 est 5
Le PGDC de 7/23 est 1

Lua[edit]

Translation of: C
function gcd(a,b)
	if b ~= 0 then
		return gcd(b, a % b)
	else
		return math.abs(a)
	end
end

function demo(a,b)
	print("GCD of " .. a .. " and " .. b .. " is " .. gcd(a, b))
end

demo(100, 5)
demo(5, 100)
demo(7, 23)

Output:

GCD of 100 and 5 is 5
GCD of 5 and 100 is 5
GCD of 7 and 23 is 1

Faster iterative solution of Euclid:

function gcd(a,b)
    while b~=0 do 
        a,b=b,a%b
    end
    return math.abs(a)
end

Lucid[edit]

dataflow algorithm[edit]

gcd(n,m) where
   z = [% n, m %] fby if x > y then [% x - y, y %] else [% x, y - x%] fi;
   x = hd(z);
   y = hd(tl(z));
   gcd(n, m) = (x asa x*y eq 0) fby eod;
end

Luck[edit]

function gcd(a: int, b: int): int = (
   if a==0 then b
   else if b==0 then a
   else if a>b then gcd(b, a % b)
   else gcd(a, b % a)
)

M2000 Interpreter[edit]

gcd=lambda (u as long, v as long) -> {
           =if(v=0&->abs(u), lambda(v, u mod v))
}
gcd_Iterative= lambda (m as long, n as long) -> {
   while m  {
       let old_m = m
       m = n mod m
       n = old_m
   }
   =abs(n)
}
Module CheckGCD (f){
      Print f(49865, 69811)=9973
      Def ExpType$(x)=Type$(x)
      Print ExpType$(f(49865, 69811))="Long"
}
CheckGCD gcd
CheckGCD gcd_Iterative

m4[edit]

This should work in any POSIX-compliant m4. I have tested it with GNU m4, OpenBSD m4, and Heirloom Devtools m4. It is Euler’s algorithm.

divert(-1)
define(`gcd',
  `ifelse(eval(`0 <= (' $1 `)'),`0',`gcd(eval(`-(' $1 `)'),eval(`(' $2 `)'))',
          eval(`0 <= (' $2 `)'),`0',`gcd(eval(`(' $1 `)'),eval(`-(' $2 `)'))',
          eval(`(' $1 `) == 0'),`0',`gcd(eval(`(' $2 `) % (' $1 `)'),eval(`(' $1 `)'))',
          eval(`(' $2 `)'))')
divert`'dnl
dnl
gcd(0, 0) = 0
gcd(24140, 40902) = 34
gcd(-24140, -40902) = 34
gcd(-40902, 24140) = 34
gcd(40902, -24140) = 34
Output:
0 = 0
34 = 34
34 = 34
34 = 34
34 = 34

Maple[edit]

To compute the greatest common divisor of two integers in Maple, use the procedure igcd.

igcd( a, b )

For example,

> igcd( 24, 15 );
                3

Mathematica / Wolfram Language[edit]

GCD[a, b]

MATLAB[edit]

function [gcdValue] = greatestcommondivisor(integer1, integer2)
   gcdValue = gcd(integer1, integer2);

Maxima[edit]

/* There is a function gcd(a, b) in Maxima, but one can rewrite it */
gcd2(a, b) := block([a: abs(a), b: abs(b)], while b # 0 do [a, b]: [b, mod(a, b)], a)$

/* both will return 2^97 * 3^48 */
gcd(100!, 6^100), factor;
gcd2(100!, 6^100), factor;

MAXScript[edit]

Iterative Euclid algorithm[edit]

fn gcdIter a b =
(
    while b > 0 do
    (
        c = mod a b
        a = b
        b = c
    )
    abs a
)

Recursive Euclid algorithm[edit]

fn gcdRec a b =
(
    if b > 0 then gcdRec b (mod a b) else abs a
)

Mercury[edit]

Recursive Euclid algorithm[edit]

:- module gcd.

:- interface.
:- import_module integer.

:- func gcd(integer, integer) = integer.

:- implementation.

:- pragma memo(gcd/2).
gcd(A, B) = (if B = integer(0) then A else gcd(B, A mod B)).

An example console program to demonstrate the gcd module:

:- module test_gcd.

:- interface.

:- import_module io.

:- pred main(io::di, io::uo) is det.

:- implementation.

:- import_module char.
:- import_module gcd.
:- import_module integer.
:- import_module list.
:- import_module string.

main(!IO) :-
    command_line_arguments(Args, !IO),
    filter(is_all_digits, Args, CleanArgs),

    Arg1 = list.det_index0(CleanArgs, 0),
    Arg2 = list.det_index0(CleanArgs, 1),
    A = integer.det_from_string(Arg1),
    B = integer.det_from_string(Arg2),

    Fmt = integer.to_string,
    GCD = gcd(A, B),
    io.format("gcd(%s, %s) = %s\n", [s(Fmt(A)), s(Fmt(B)), s(Fmt(GCD))], !IO).

Example output:

gcd(70000000000000000000000, 60000000000000000000000000) = 10000000000000000000000

MINIL[edit]

// Greatest common divisor
00 0E  GCD:   ENT  R0
01 1E         ENT  R1
02 21  Again: R2 = R1
03 10  Loop:  R1 = R0
04 02         R0 = R2
05 2D  Minus: DEC  R2
06 8A         JZ   Stop
07 1D         DEC  R1
08 C5         JNZ  Minus
09 83         JZ   Loop
0A 1D  Stop:  DEC  R1
0B C2         JNZ  Again
0C 80         JZ   GCD   // Display GCD in R0

MiniScript[edit]

Using an iterative Euclidean algorithm:

gcd = function(a, b)
    while b
        temp = b
        b = a % b
        a = temp
    end while
    return abs(a)
end function

print gcd(18,12)
Output:
6

MiniZinc[edit]

function var int: gcd(int:a2,int:b2) =
  let {
    int:a1 = max(a2,b2);
    int:b1 = min(a2,b2);
    array[0..a1,0..b1] of var int: gcd;
    constraint forall(a in 0..a1)(
      forall(b in 0..b1)(
        gcd[a,b] ==
        if (b == 0) then
          a
        else
          gcd[b, a mod b]
        endif
      )
    )
  } in gcd[a1,b1];  
 
var int: gcd1 = gcd(8,12);
solve satisfy;
output [show(gcd1),"\n"];
Output:
6

MIPS Assembly[edit]

gcd:
  # a0 and a1 are the two integer parameters
  # return value is in v0
  move $t0, $a0
  move $t1, $a1
loop:
  beq $t1, $0, done
  div $t0, $t1
  move $t0, $t1
  mfhi $t1
  j loop
done:
  move $v0, $t0
  jr $ra

МК-61/52[edit]

ИПA	ИПB	/	П9	КИП9	ИПA	ИПB	ПA	ИП9	*
-	ПB	x=0	00	ИПA	С/П

Enter: n = РA, m = РB (n > m).

ML[edit]

mLite[edit]

fun gcd (a, 0) = a
      | (0, b) = b
      | (a, b) where (a < b)
               = gcd (a, b rem a)
      | (a, b) = gcd (b, a rem b)

ML / Standard ML[edit]

See also #Standard ML.

fun gcd a 0 = a
  | gcd a b = gcd b (a mod b)

Modula-2[edit]

MODULE ggTkgV;

FROM    InOut           IMPORT  ReadCard, WriteCard, WriteLn, WriteString, WriteBf;

VAR   x, y, u, v        : CARDINAL;

BEGIN
  WriteString ("x = ");         WriteBf;        ReadCard (x);
  WriteString ("y = ");         WriteBf;        ReadCard (y);
  u := x;
  v := y;
  WHILE  x # y  DO
    (*  ggT (x, y) = ggT (x0, y0), x * v + y * u = 2 * x0 * y0          *)
    IF  x > y  THEN
      x := x - y;
      u := u + v
    ELSE
      y := y - x;
      v := v + u
    END
  END;
  WriteString ("ggT =");        WriteCard (x, 6);               WriteLn;
  WriteString ("kgV =");        WriteCard ((u+v) DIV 2, 6);     WriteLn;
  WriteString ("u =");          WriteCard (u, 6);               WriteLn;
  WriteString ("v =");          WriteCard (v , 6);              WriteLn
END ggTkgV.

Producing the output

jan@Beryllium:~/modula/Wirth/PIM$ ggtkgv
x = 12
y = 20
ggT =     4
kgV =    60
u =    44
v =    76
jan@Beryllium:~/modula/Wirth/PIM$ ggtkgv
x = 123
y = 255
ggT =     3
kgV = 10455
u = 13773
v =  7137

Modula-3[edit]

MODULE GCD EXPORTS Main;

IMPORT IO, Fmt;

PROCEDURE GCD(a, b: CARDINAL): CARDINAL =
  BEGIN
    IF a = 0 THEN
      RETURN b;
    ELSIF b = 0 THEN
      RETURN a;
    ELSIF a > b THEN
      RETURN GCD(b, a MOD b);
    ELSE
      RETURN GCD(a, b MOD a);
    END;
  END GCD;

BEGIN
  IO.Put("GCD of 100, 5 is " & Fmt.Int(GCD(100, 5)) & "\n");
  IO.Put("GCD of 5, 100 is " & Fmt.Int(GCD(5, 100)) & "\n");
  IO.Put("GCD of 7, 23 is " & Fmt.Int(GCD(7, 23)) & "\n");
END GCD.

Output:

GCD of 100, 5 is 5
GCD of 5, 100 is 5
GCD of 7, 23 is 1

Monicelli[edit]

 #main function (needs two ints from stdin)
Lei ha clacsonato bitumatissimi cari amici ospiti
voglio arrivata, Necchi bitumati qui tra noi benvenuti ENERGIAAh
voglio espertizzata, Necchi
mi porga arrivata mi porga espertizzata bitumata mi raccomando
voglio garantita, Necchi come se fosse prematurata la supercazzola accanita con arrivata, espertizzata o scherziamo?
brematurata la supercazzola novella o scherziamo? bitumata ma dai
garantita a posterdati
# gcd function
blinda la supercazzola Necchi accanita con visibilio Necchi, sgomento Necchi o scherziamo?
voglio la catarsi, Necchi bituma, e come bituma lui non bituma nessuno
voglio l'entusiasmo, Necchi bituma, chi di bituma vive bitumato muore
che cos'è il visibilio? sgomento: vaffanzum visibilio!
o magari maggiore di sgomento: catarsi come fosse visibilio meno sgomento bitumato ma non troppo
entusiasmo come fosse sgomento bitumato anche piu del necessario
o tarapia tapioco: catarsi come fosse sgomento meno il visibilio bitumante dai tempi andati
entusiasmo come fosse visibilio e velocità di esecuzione
voglio la spensierataggine, Necchi come fosse prematurata la supercazzola accanita con catarsi, entusiasmo o scherziamo?
vaffanzum la spensierataggine!
# prints new line
blinda la supercazzola novella o scherziamo?
voglio novita, Mascetti come se fosse 10 bituma come fosse una lungaggine, uno scherzo di mano
novita a posterdati!

MUMPS[edit]

GCD(A,B)
 QUIT:((A/1)'=(A\1))!((B/1)'=(B\1)) 0
 SET:A<0 A=-A
 SET:B<0 B=-B
 IF B'=0
 FOR  SET T=A#B,A=B,B=T QUIT:B=0 ;ARGUEMENTLESS FOR NEEDS TWO SPACES
 QUIT A

Ouput:

CACHE>S X=$$GCD^ROSETTA(12,24) W X
12
CACHE>S X=$$GCD^ROSETTA(24,-112) W X
8
CACHE>S X=$$GCD^ROSETTA(24,-112.2) W X
0

MySQL[edit]

DROP FUNCTION IF EXISTS gcd;
DELIMITER |

CREATE FUNCTION gcd(x INT, y INT)
RETURNS INT
BEGIN
  SET @dividend=GREATEST(ABS(x),ABS(y));
  SET @divisor=LEAST(ABS(x),ABS(y));
  IF @divisor=0 THEN
    RETURN @dividend;
  END IF;
  SET @gcd=NULL;
  SELECT gcd INTO @gcd FROM
    (SELECT @tmp:=@dividend,
            @dividend:=@divisor AS gcd,
            @divisor:=@tmp % @divisor AS remainder
       FROM mysql.help_relation WHERE @divisor>0) AS x
    WHERE remainder=0;
  RETURN @gcd;
END;|

DELIMITER ;

SELECT gcd(12345, 9876);
+------------------+
| gcd(12345, 9876) |
+------------------+
|             2469 |
+------------------+
1 row in set (0.00 sec)

Nanoquery[edit]

Translation of: Java

Iterative[edit]

def gcd(a, b)
	factor = a.min(b)

	for loop in range(factor, 2)
		if (a % loop = 0) and (b % loop = 0)
			return loop
		end
	end

	return 1
end

Iterative Euclid's Method[edit]

def gcd_euclid(a, b)
	while b > 0
		c = a % b
		a = b
		b = c
	end
	return a
end

NetRexx[edit]

/* NetRexx */
options replace format comments java crossref symbols nobinary

numeric digits 2000
runSample(arg)
return

-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
-- Euclid's algorithm - iterative implementation
method gcdEucidI(a_, b_) public static
  loop while b_ > 0
    c_ = a_ // b_
    a_ = b_
    b_ = c_
    end
  return a_

-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
-- Euclid's algorithm - recursive implementation
method gcdEucidR(a_, b_) public static
  if b_ \= 0 then a_ = gcdEucidR(b_, a_ // b_)
  return a_

-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
method runSample(arg) private static
  -- pairs of numbers, each number in the pair separated by a colon, each pair separated by a comma
  parse arg tests
  if tests = '' then
    tests = '0:0, 6:4, 7:21, 12:36, 33:77, 41:47, 99:51, 100:5, 7:23, 1989:867, 12345:9876, 40902:24140, 49865:69811, 137438691328:2305843008139952128'

  -- most of what follows is for formatting
  xiterate = 0
  xrecurse = 0
  ll_ = 0
  lr_ = 0
  lgi = 0
  lgr = 0
  loop i_ = 1 until tests = ''
    xiterate[0] = i_
    xrecurse[0] = i_
    parse tests pair ',' tests
    parse pair l_ ':' r_ .

    -- get the GCDs
    gcdi = gcdEucidI(l_, r_)
    gcdr = gcdEucidR(l_, r_)

    xiterate[i_] = l_ r_ gcdi
    xrecurse[i_] = l_ r_ gcdr
    ll_ = ll_.max(l_.strip.length)
    lr_ = lr_.max(r_.strip.length)
    lgi = lgi.max(gcdi.strip.length)
    lgr = lgr.max(gcdr.strip.length)
    end i_
  -- save formatter sizes in stems
  xiterate[-1] = ll_ lr_ lgi
  xrecurse[-1] = ll_ lr_ lgr

  -- present results
  showResults(xiterate, 'Euclid''s algorithm - iterative')
  showResults(xrecurse, 'Euclid''s algorithm - recursive')
  say
  if verifyResults(xiterate, xrecurse) then
    say 'Success: Results of iterative and recursive methods match'
  else
    say 'Error:   Results of iterative and recursive methods do not match'
  say
  return

-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
method showResults(stem, title) public static
  say
  say title
  parse stem[-1] ll lr lg
  loop v_ = 1 to stem[0]
    parse stem[v_] lv rv gcd .
    say lv.right(ll)',' rv.right(lr) ':' gcd.right(lg)
    end v_
  return

-- 09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)09:53, 27 August 2022 (UTC)~~
method verifyResults(stem1, stem2) public static returns boolean
  if stem1[0] \= stem2[0] then signal BadArgumentException
  T = (1 == 1)
  F = \T
  verified = T
  loop i_ = 1 to stem1[0]
    if stem1[i_] \= stem2[i_] then do
      verified = F
      leave i_
      end
    end i_
  return verified
Output:
Euclid's algorithm - iterative
           0,                   0 :      0
           6,                   4 :      2
           7,                  21 :      7
          12,                  36 :     12
          33,                  77 :     11
          41,                  47 :      1
          99,                  51 :      3
         100,                   5 :      5
           7,                  23 :      1
        1989,                 867 :     51
       12345,                9876 :   2469
       40902,               24140 :     34
       49865,               69811 :   9973
137438691328, 2305843008139952128 : 262144

Euclid's algorithm - recursive
           0,                   0 :      0
           6,                   4 :      2
           7,                  21 :      7
          12,                  36 :     12
          33,                  77 :     11
          41,                  47 :      1
          99,                  51 :      3
         100,                   5 :      5
           7,                  23 :      1
        1989,                 867 :     51
       12345,                9876 :   2469
       40902,               24140 :     34
       49865,               69811 :   9973
137438691328, 2305843008139952128 : 262144

Success: Results of iterative and recursive methods match

NewLISP[edit]

(gcd 12 36)  
   12

Nial[edit]

Nial provides gcd in the standard lib.

|loaddefs 'niallib/gcd.ndf'
|gcd 6 4
=2

defining it for arrays

# red is the reduction operator for a sorted list
# one is termination condition
red is cull filter (0 unequal) link [mod [rest, first] , first]
one is or [= [1 first, tally], > [2 first,  first]]
gcd is fork [one, first, gcd red] sort <=

Using it

|gcd 9 6 3
=3

Nim[edit]

Translation of: Pascal

Recursive Euclid algorithm[edit]

func gcd_recursive*(u, v: SomeSignedInt): int64 =
    if u mod v != 0:
        result = gcd_recursive(v, u mod v)
    else:
        result = abs(v)

when isMainModule:
  import strformat
  let (x, y) = (49865, 69811)
  echo &"gcd({x}, {y}) = {gcd_recursive(49865, 69811)}"
Output:
gcd(49865, 69811) = 9973

Iterative Euclid algorithm[edit]

func gcd_iterative*(u, v: SomeSignedInt): int64 =
  var u = u
  var v = v
  while v != 0:
      u = u mod v
      swap u, v
  result = abs(u)

when isMainModule:
  import strformat
  let (x, y) = (49865, 69811)
  echo &"gcd({x}, {y}) = {gcd_iterative(49865, 69811)}")
Output:
gcd(49865, 69811) = 9973

Iterative binary algorithm[edit]

template isEven(n: int64): bool = (n and 1) == 0

func gcd_binary*(u, v: int64): int64 =

  var u = abs(u)
  var v = abs(v)
  if u < v: swap u, v

  if v == 0: return u

  var k = 1
  while u.isEven and v.isEven:
    u = u shr 1
    v = v shr 1
    k = k shl 1
  var t = if u.isEven: u else: -v
  while t != 0:
    while t.isEven: t = ashr(t, 1)
    if t > 0: u = t
    else: v = -t
    t = u - v
  result = u * k

when isMainModule:
  import strformat
  let (x, y) = (49865, 69811)
  echo &"gcd({x}, {y}) = {gcd_binary(49865, 69811)}"
Output:
gcd(49865, 69811) = 9973

Oberon-2[edit]

Works with oo2c version 2

MODULE GCD;
(* Greatest Common Divisor *)
IMPORT 
  Out;
  
  PROCEDURE Gcd(a,b: LONGINT):LONGINT;
  VAR
    r: LONGINT;
  BEGIN
    LOOP
      r := a MOD b;
      IF r = 0 THEN RETURN b END;
      a := b;b := r
    END
  END Gcd;
BEGIN
  Out.String("GCD of    12 and     8 : ");Out.LongInt(Gcd(12,8),4);Out.Ln;
  Out.String("GCD of   100 and     5 : ");Out.LongInt(Gcd(100,5),4);Out.Ln;
  Out.String("GCD of     7 and    23 : ");Out.LongInt(Gcd(7,23),4);Out.Ln;
  Out.String("GCD of    24 and  -112 : ");Out.LongInt(Gcd(12,8),4);Out.Ln;
  Out.String("GCD of 40902 and 24140 : ");Out.LongInt(Gcd(40902,24140),4);Out.Ln
END GCD.

Output:

GCD of    12 and     8 :    4
GCD of   100 and     5 :    5
GCD of     7 and    23 :    1
GCD of    24 and  -112 :    4
GCD of 40902 and 24140 :   34

Objeck[edit]

bundle Default {
  class GDC {
    function : Main(args : String[]), Nil {
      for(x := 1; x < 36; x += 1;) {
        IO.Console->GetInstance()->Print("GCD of ")->Print(36)->Print(" and ")->Print(x)->Print(" is ")->PrintLine(GDC(36, x));
      };
    }
    
    function : native : GDC(a : Int, b : Int), Int {
      t : Int;
      
      if(a > b) {
        t := b;  b := a;  a := t;
      };
     
      while (b <> 0) {
        t := a % b;  a := b;  b := t;
      };
      
      return a;
    }
  }
}

OCaml[edit]

let rec gcd a b =
  if      a = 0 then b
  else if b = 0 then a
  else if a > b then gcd b (a mod b)
  else               gcd a (b mod a)

A little more idiomatic version:

let rec gcd1 a b =
  match (a mod b) with
    0 -> b
  | r -> gcd1 b r

Built-in[edit]

#load "nums.cma";;
open Big_int;;
let gcd a b =
  int_of_big_int (gcd_big_int (big_int_of_int a) (big_int_of_int b))

Octave[edit]

r = gcd(a, b)

Oforth[edit]

gcd is already defined into Integer class :

128 96 gcd

Source of this method is (see Integer.of file) :

Integer method: gcd  self while ( dup ) [ tuck mod ] drop ;

Ol[edit]

(print (gcd 1071 1029))
; ==> 21

Order[edit]

Translation of: bc
#include <order/interpreter.h>

#define ORDER_PP_DEF_8gcd ORDER_PP_FN( \
8fn(8U, 8V,                            \
    8if(8isnt_0(8V), 8gcd(8V, 8remainder(8U, 8V)), 8U)))
// No support for negative numbers

Oz[edit]

declare
  fun {UnsafeGCD A B}
     if B == 0 then
        A
     else
        {UnsafeGCD B A mod B}
     end
  end

  fun {GCD A B}
     if A == 0 andthen B == 0 then
        raise undefined(gcd 0 0) end
     else
        {UnsafeGCD {Abs A} {Abs B}}
     end
  end
in
  {Show {GCD 456 ~632}}

PARI/GP[edit]

gcd(a,b)

PASCAL program GCF (INPUT, OUTPUT);

 var
   a,b,c:integer;
 begin
   writeln('Enter 1st number');
   read(a);
   writeln('Enter 2nd number');
   read(b);
   while (a*b<>0)
     do
     begin
       c:=a;
       a:=b mod a;
       b:=c;
     end;
   writeln('GCF :=', a+b );
 end.

By: NG

Pascal / Delphi / Free Pascal[edit]

Recursive Euclid algorithm[edit]

Works with: Free Pascal version version 3.2.0
PROGRAM EXRECURGCD.PAS;

{$IFDEF FPC}
    {$mode objfpc}{$H+}{$J-}{R+}
{$ELSE}
    {$APPTYPE CONSOLE}
{$ENDIF}

(*) 
    Free Pascal Compiler version 3.2.0 [2020/06/14] for x86_64
    The free and readable alternative at C/C++ speeds
    compiles natively to almost any platform, including raspberry PI
(*)

FUNCTION gcd_recursive(u, v: longint): longint;

    BEGIN
        IF ( v = 0 ) THEN Exit ( u ) ;
        result := gcd_recursive ( v, u MOD v ) ;
    END;

BEGIN

    WriteLn ( gcd_recursive ( 231, 7 ) ) ;

END.
JPD 2021/03/14

Iterative Euclid algorithm[edit]

function gcd_iterative(u, v: longint): longint;
  var
    t: longint;
  begin
    while v <> 0 do
    begin
      t := u;
      u := v;
      v := t mod v;
    end;
    gcd_iterative := abs(u);
  end;

Iterative binary algorithm[edit]

function gcd_binary(u, v: longint): longint;
  var
    t, k: longint;
  begin
    u := abs(u);
    v := abs(v); 
    if u < v then
    begin
      t := u;
      u := v;
      v := t;
    end;
    if v = 0 then
      gcd_binary := u
    else
    begin
      k := 1;
      while (u mod 2 = 0) and (v mod 2 = 0) do
      begin
        u := u >> 1;
        v := v >> 1;
	k := k << 1;
      end;
      if u mod 2 = 0 then
        t := u
      else
        t := -v;
      while t <> 0 do
      begin
        while t mod 2 = 0 do
          t := t div 2;
        if t > 0 then
          u := t
        else
          v := -t;
        t := u - v;
      end;
      gcd_binary := u * k;
    end;
  end;

Demo program:

Program GreatestCommonDivisorDemo(output);
begin
  writeln ('GCD(', 49865, ', ', 69811, '): ', gcd_iterative(49865, 69811), ' (iterative)');
  writeln ('GCD(', 49865, ', ', 69811, '): ', gcd_recursive(49865, 69811), ' (recursive)');
  writeln ('GCD(', 49865, ', ', 69811, '): ', gcd_binary   (49865, 69811), ' (binary)');
end.

Output:

GCD(49865, 69811): 9973 (iterative)
GCD(49865, 69811): 9973 (recursive)
GCD(49865, 69811): 9973 (binary)

Perl[edit]

Iterative Euclid algorithm[edit]

sub gcd_iter($$) {
  my ($u, $v) = @_;
  while ($v) {
    ($u, $v) = ($v, $u % $v);
  }
  return abs($u);
}

Recursive Euclid algorithm[edit]

sub gcd($$) {
  my ($u, $v) = @_;
  if ($v) {
    return gcd($v, $u % $v);
  } else {
    return abs($u);
  }
}

Iterative binary algorithm[edit]

sub gcd_bin($$) {
  my ($u, $v) = @_;
  $u = abs($u);
  $v = abs($v);
  if ($u < $v) {
    ($u, $v) = ($v, $u);
  }
  if ($v == 0) {
    return $u;
  }
  my $k = 1;
  while ($u & 1 == 0 && $v & 1 == 0) {
    $u >>= 1;
    $v >>= 1;
    $k <<= 1;
  }
  my $t = ($u & 1) ? -$v : $u;
  while ($t) {
    while ($t & 1 == 0) {
      $t >>= 1;
    }
    if ($t > 0) {
      $u = $t;
    } else {
      $v = -$t;
    }
    $t = $u - $v;
  }
  return $u * $k;
}

Modules[edit]

All three modules will take large integers as input, e.g. gcd("68095260063025322303723429387", "51306142182612010300800963053"). Other possibilities are Math::Cephes euclid, Math::GMPz gcd and gcd_ui.

# Fastest, takes multiple inputs
use Math::Prime::Util "gcd";
$gcd = gcd(49865, 69811);

# In CORE.  Slowest, takes multiple inputs, result is a Math::BigInt unless converted
use Math::BigInt;
$gcd = Math::BigInt::bgcd(49865, 69811)->numify;

# Result is a Math::Pari object unless converted
use Math::Pari "gcd";
$gcd = gcd(49865, 69811)->pari2iv

Notes on performance[edit]

use Benchmark qw(cmpthese);
use Math::BigInt;
use Math::Pari;
use Math::Prime::Util;

my $u = 40902;
my $v = 24140;
cmpthese(-5, {
  'gcd_rec' => sub { gcd($u, $v); },
  'gcd_iter' => sub { gcd_iter($u, $v); },
  'gcd_bin' => sub { gcd_bin($u, $v); },
  'gcd_bigint' => sub { Math::BigInt::bgcd($u,$v)->numify(); },
  'gcd_pari' => sub { Math::Pari::gcd($u,$v)->pari2iv(); },
  'gcd_mpu' => sub { Math::Prime::Util::gcd($u,$v); },
});

Output on 'Intel i3930k 4.2GHz' / Linux / Perl 5.20:

                Rate gcd_bigint   gcd_bin   gcd_rec  gcd_iter gcd_pari   gcd_mpu
gcd_bigint   39939/s         --      -83%      -94%      -95%     -98%      -99%
gcd_bin     234790/s       488%        --      -62%      -70%     -88%      -97%
gcd_rec     614750/s      1439%      162%        --      -23%     -68%      -91%
gcd_iter    793422/s      1887%      238%       29%        --     -58%      -89%
gcd_pari   1896544/s      4649%      708%      209%      139%       --      -73%
gcd_mpu    7114798/s     17714%     2930%     1057%      797%     275%        --

Phix[edit]

result is always positive, except for gcd(0,0) which is 0
atom parameters allow greater precision, but any fractional parts are immediately and deliberately discarded.
Actually, it is an autoinclude, reproduced below. The first parameter can be a sequence, in which case the second parameter (if provided) is ignored.

function gcd(object u, atom v=0)
atom t
    if sequence(u) then
        v = u[1]                        -- (for the typecheck)
        t = floor(abs(v))
        for i=2 to length(u) do
            v = u[i]                    -- (for the typecheck)
            t = gcd(t,v)
        end for
        return t
    end if
    u = floor(abs(u))
    v = floor(abs(v))
    while v do
        t = u
        u = v
        v = remainder(t, v)
    end while
    return u
end function

Sample results

?gcd(0,0)           -- 0
?gcd(24,-112)       -- 8
?gcd(0, 10)         -- 10
?gcd(10, 0)         -- 10
?gcd(-10, 0)        -- 10
?gcd(0, -10)        -- 10
?gcd(9, 6)          -- 3
?gcd(6, 9)          -- 3
?gcd(-6, 9)         -- 3
?gcd(9, -6)         -- 3
?gcd(6, -9)         -- 3
?gcd(-9, 6)         -- 3
?gcd(40902, 24140)  -- 34   
printf(1,"%d\n",gcd(70000000000000000000, 
                    60000000000000000000000))
                --  10000000000000000000
?gcd({57,0,-45,-18,90,447}) -- 3

PHP[edit]

Iterative[edit]

function gcdIter($n, $m) {
    while(true) {
        if($n == $m) {
            return $m;
        }
        if($n > $m) {
            $n -= $m;
        } else {
            $m -= $n;
        }
    }
}

Recursive[edit]

function gcdRec($n, $m)
{
    if($m > 0)
        return gcdRec($m, $n % $m);
    else
        return abs($n);
}

PicoLisp[edit]

(de gcd (A B)
   (until (=0 B)
      (let M (% A B)
         (setq A B B M) ) )
   (abs A) )

PL/I[edit]

GCD: procedure (a, b) returns (fixed binary (31)) recursive;
   declare (a, b) fixed binary (31);

   if b = 0 then return (a);

   return (GCD (b, mod(a, b)) );

end GCD;

Pop11[edit]

Built-in gcd[edit]

gcd_n(15, 12, 2) =>

Note: the last argument gives the number of other arguments (in this case 2).

Iterative Euclid algorithm[edit]

define gcd(k, l) -> r;
    lvars k , l, r = l;
    abs(k) -> k;
    abs(l) -> l;
    if k < l then (k, l) -> (l, k) endif;
    while l /= 0 do
        (l, k rem l) -> (k, l)
    endwhile;
    k -> r;
enddefine;

PostScript[edit]

Library: initlib
/gcd {
{   
    {0 gt} {dup rup mod} {pop exit} ifte
} loop
}.

With no external lib, recursive

/gcd {
   dup 0 ne {
      dup 3 1 roll mod gcd
   } { pop } ifelse
} def

PowerShell[edit]

Recursive Euclid Algorithm[edit]

function Get-GCD ($x, $y)
{
  if ($x -eq $y) { return $y }
  if ($x -gt $y) {
    $a = $x
    $b = $y
  }
  else {
    $a = $y
    $b = $x
  }
  while ($a % $b -ne 0) {
    $tmp = $a % $b
    $a = $b
    $b = $tmp
  }
  return $b
}

or shorter (taken from Python implementation)

function Get-GCD ($x, $y) {
  if ($y -eq 0) { $x } else { Get-GCD $y ($x%$y) }
}

Iterative Euclid Algorithm[edit]

based on Python implementation

Function Get-GCD( $x, $y ) {
    while ($y -ne 0) {
        $x, $y = $y, ($x % $y)
    }
    [Math]::abs($x)
}

Prolog[edit]

Recursive Euclid Algorithm[edit]

gcd(X, 0, X):- !.
gcd(0, X, X):- !.
gcd(X, Y, D):- X > Y, !, Z is X mod Y, gcd(Y, Z, D).
gcd(X, Y, D):- Z is Y mod X, gcd(X, Z, D).

Repeated Subtraction[edit]

gcd(X, 0, X):- !.
gcd(0, X, X):- !.
gcd(X, Y, D):- X =< Y, !, Z is Y - X, gcd(X, Z, D).
gcd(X, Y, D):- gcd(Y, X, D).

Purity[edit]

data Iterate = f => FoldNat <const id, g => $g . $f>

data Sub = Iterate Pred
data IsZero = <const True, const False> . UnNat

data Eq = FoldNat 
          <
              const IsZero, 
              eq => n => IfThenElse (IsZero $n) 
                         False 
                         ($eq (Pred $n))
          >

data step = gcd => n => m => 
                    IfThenElse (Eq $m $n) 
                        (Pair $m $n) 
                        (IfThenElse (Compare Leq $n $m) 
                            ($gcd (Sub $m $n) $m) 
                            ($gcd (Sub $n $m) $n))

data gcd = Iterate (gcd => uncurry (step (curry $gcd)))

Python[edit]

Built-in[edit]

Works with: Python version 2.6+
from fractions import gcd
Works with: Python version 3.7

(Note that fractions.gcd is now deprecated in Python 3)

from math import gcd

Iterative Euclid algorithm[edit]

def gcd_iter(u, v):
    while v:
        u, v = v, u % v
    return abs(u)

Recursive Euclid algorithm[edit]

Interpreter: Python 2.5

def gcd(u, v):
    return gcd(v, u % v) if v else abs(u)

Tests[edit]

>>> gcd(0,0)
0
>>> gcd(0, 10) == gcd(10, 0) == gcd(-10, 0) == gcd(0, -10) == 10
True
>>> gcd(9, 6) == gcd(6, 9) == gcd(-6, 9) == gcd(9, -6) == gcd(6, -9) == gcd(-9, 6) == 3
True
>>> gcd(8, 45) == gcd(45, 8) == gcd(-45, 8) == gcd(8, -45) == gcd(-8, 45) == gcd(45, -8) == 1
True
>>> gcd(40902, 24140) # check Knuth :)
34

Iterative binary algorithm[edit]

See The Art of Computer Programming by Knuth (Vol.2)

def gcd_bin(u, v):
    u, v = abs(u), abs(v) # u >= 0, v >= 0
    if u < v:
        u, v = v, u # u >= v >= 0
    if v == 0:
        return u
   
    # u >= v > 0
    k = 1
    while u & 1 == 0 and v & 1 == 0: # u, v - even
        u >>= 1; v >>= 1
        k <<= 1
       
    t = -v if u & 1 else u
    while t:
        while t & 1 == 0:
            t >>= 1
        if t > 0:
            u = t
        else:
            v = -t
        t = u - v
    return u * k

Notes on performance[edit]

gcd(40902, 24140) takes us about 17 µsec (Euclid, not built-in)

gcd_iter(40902, 24140) takes us about 11 µsec

gcd_bin(40902, 24140) takes us about 41 µsec

Qi[edit]

(define gcd
  A 0 -> A
  A B -> (gcd B (MOD A B)))

Quackery[edit]

  [ [ dup while
      tuck mod again ]
    drop abs ]         is gcd ( n n --> n )

R[edit]

Recursive:

"%gcd%" <- function(u, v) {
  ifelse(u %% v != 0, v %gcd% (u%%v), v)
}

Iterative:

"%gcd%" <- function(v, t) {
  while ( (c <- v%%t) != 0 ) {
    v <- t
    t <- c
  }
  t
}
Output:

Same either way.

> print(50 %gcd% 75)
[1] 25

Racket[edit]

Racket provides a built-in gcd function. Here's a program that computes the gcd of 14 and 63:

#lang racket

(gcd 14 63)

Here's an explicit implementation. Note that since Racket is tail-calling, the memory behavior of this program is "loop-like", in the sense that this program will consume no more memory than a loop-based implementation.

#lang racket

;; given two nonnegative integers, produces their greatest 
;; common divisor using Euclid's algorithm
(define (gcd a b)
  (if (= b 0)
      a
      (gcd b (modulo a b))))

;; some test cases!
(module+ test
  (require rackunit)
  (check-equal? (gcd (* 2 3 3 7 7)
                     (* 3 3 7 11))
                (* 3 3 7))
  (check-equal? (gcd 0 14) 14)
  (check-equal? (gcd 13 0) 13))

Raku[edit]

(formerly Perl 6)

Iterative[edit]

sub gcd (Int $a is copy, Int $b is copy) {
   $a & $b == 0 and fail;
   ($a, $b) = ($b, $a % $b) while $b;
   return abs $a;
}

Recursive[edit]

multi gcd (0,      0)      { fail }
multi gcd (Int $a, 0)      { abs $a }
multi gcd (Int $a, Int $b) { gcd $b, $a % $b }

Concise[edit]

my &gcd = { ($^a.abs, $^b.abs, * % * ... 0)[*-2] }

Actually, it's a built-in infix[edit]

my $gcd = $a gcd $b;

Because it's an infix, you can use it with various meta-operators:

[gcd] @list;         # reduce with gcd
@alist Zgcd @blist;  # lazy zip with gcd
@alist Xgcd @blist;  # lazy cross with gcd
@alist »gcd« @blist; # parallel gcd

Rascal[edit]

Iterative Euclidean algorithm[edit]

public int gcd_iterative(int a, b){
	if(a == 0) return b;
	while(b != 0){
		if(a > b) a -= b;
		else b -= a;}
	return a;
}

An example:

rascal>gcd_iterative(1989, 867)
int: 51

Recursive Euclidean algorithm[edit]

public int gcd_recursive(int a, b){
	return (b == 0) ? a : gcd_recursive(b, a%b);
}

An example:

rascal>gcd_recursive(1989, 867)
int: 51

Raven[edit]

Recursive Euclidean algorithm[edit]

define gcd use $u, $v
   $v 0 > if
      $u $v %   $v  gcd
   else
      $u abs

24140 40902 gcd
Output:
34

REBOL[edit]

gcd: func [
    {Returns the greatest common divisor of m and n.}
    m [integer!]
    n [integer!]
    /local k
] [
    ; Euclid's algorithm
    while [n > 0] [
        k: m
        m: n
        n: k // m
    ]
    m
]

Retro[edit]

This is from the math extensions library.

: gcd ( ab-n ) [ tuck mod dup ] while drop ;

REXX[edit]

version 1[edit]

The GCD subroutine can handle any number of arguments,   it can also handle any number of integers within any
argument(s),   making it easier to use when computing Frobenius numbers   (also known as   postage stamp   or  
coin   numbers).

/*REXX program calculates the  GCD (Greatest Common Divisor)  of any number of integers.*/
numeric digits 2000                              /*handle up to 2k decimal dig integers.*/
call gcd 0 0            ;    call gcd 55 0     ;       call gcd 0    66
call gcd 7,21           ;    call gcd 41,47    ;       call gcd 99 , 51
call gcd 24, -8         ;    call gcd -36, 9   ;       call gcd -54, -6
call gcd 14 0 7         ;    call gcd 14 7 0   ;       call gcd 0  14 7
call gcd 15 10 20 30 55 ;    call gcd 137438691328  2305843008139952128 /*◄──2 perfect#s*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
gcd: procedure;  $=;              do i=1 for  arg();  $=$ arg(i);  end       /*arg list.*/
     parse var $ x z .;  if x=0  then x=z;   x=abs(x)                        /* 0 case? */

        do j=2  to words($);   y=abs(word($,j));       if y=0  then iterate  /*is zero? */
              do until _==0;  _=x//y;  x=y;  y=_;  end /* ◄────────── the heavy lifting.*/
        end   /*j*/

     say 'GCD (Greatest Common Divisor) of '   translate(space($),",",' ')   "  is  "   x
     return x

output

GCD (Greatest Common Divisor) of  0,0   is   0
GCD (Greatest Common Divisor) of  55,0   is   55
GCD (Greatest Common Divisor) of  0,66   is   66
GCD (Greatest Common Divisor) of  7,21   is   7
GCD (Greatest Common Divisor) of  41,47   is   1
GCD (Greatest Common Divisor) of  99,51   is   3
GCD (Greatest Common Divisor) of  24,-8   is   8
GCD (Greatest Common Divisor) of  -36,9   is   9
GCD (Greatest Common Divisor) of  -54,-6   is   6
GCD (Greatest Common Divisor) of  14,0,7   is   7
GCD (Greatest Common Divisor) of  14,7,0   is   7
GCD (Greatest Common Divisor) of  0,14,7   is   7
GCD (Greatest Common Divisor) of  15,10,20,30,55   is   5
GCD (Greatest Common Divisor) of  137438691328,2305843008139952128   is   262144

version 2[edit]

Recursive function (as in PL/I):

/* REXX ***************************************************************
* using PL/I code extended to many arguments
* 17.08.2012 Walter Pachl
* 18.08.2012 gcd(0,0)=0
**********************************************************************/
numeric digits 300                  /*handle up to 300 digit numbers.*/
Call test  7,21     ,'7 '
Call test  4,7      ,'1 '
Call test 24,-8     ,'8'
Call test 55,0      ,'55'
Call test 99,15     ,'3 '
Call test 15,10,20,30,55,'5'
Call test 496,8128  ,'16'
Call test 496,8128  ,'8'            /* test wrong expectation        */
Call test 0,0       ,'0'            /* by definition                 */
Exit

test:
/**********************************************************************
* Test the gcd function
**********************************************************************/
n=arg()                             /* Number of arguments           */
gcde=arg(n)                         /* Expected result               */
gcdx=gcd(arg(1),arg(2))             /* gcd of the first 2 numbers    */
Do i=2 To n-2                       /* proceed with all the others   */
  If arg(i+1)<>0 Then   
    gcdx=gcd(gcdx,arg(i+1))
  End
If gcdx=arg(arg()) Then             /* result is as expected         */
  tag='as expected'
Else                                /* result is not correct         */
  Tag='*** wrong. expected:' gcde
numbers=arg(1)                      /* build string to show the input*/
Do i=2 To n-1
  numbers=numbers 'and' arg(i)
  End
say left('the GCD of' numbers 'is',45) right(gcdx,3) tag
Return

GCD: procedure
/**********************************************************************
* Recursive procedure as shown in PL/I
**********************************************************************/
Parse Arg a,b
if b = 0 then return abs(a)
return GCD(b,a//b)

Output:

the GCD of 7 and 21 is                          7 as expected              
the GCD of 4 and 7 is                           1 as expected              
the GCD of 24 and -8 is                         8 as expected              
the GCD of 55 and 0 is                         55 as expected              
the GCD of 99 and 15 is                         3 as expected              
the GCD of 15 and 10 and 20 and 30 and 55 is    5 as expected              
the GCD of 496 and 8128 is                     16 as expected              
the GCD of 496 and 8128 is                     16 *** wrong. expected: 8  
the GCD of 0 and 0 is                           0 as expected   

version 3[edit]

Translation of: REXX
using different argument handling-

Use as gcd(a,b,c,---) Considerably faster than version 1 (and version 2)
See http://rosettacode.org/wiki/Least_common_multiple#REXX for reasoning.

gcd: procedure
x=abs(arg(1))
do j=2 to arg()
  y=abs(arg(j))
  If y<>0 Then Do
    do until z==0
      z=x//y
      x=y
      y=z
      end
    end
  end
return x

Ring[edit]

see gcd (24, 32)
func gcd gcd, b
     while b
           c   = gcd
           gcd = b
           b   = c % b
     end
     return gcd

Ruby[edit]

That is already available as the gcd method of integers:

40902.gcd(24140)  # => 34

Here's an implementation:

def gcd(u, v)
  u, v = u.abs, v.abs
  while v > 0
    u, v = v, u % v
  end
  u
end

Rust[edit]

num crate[edit]

extern crate num;
use num::integer::gcd;

Iterative Euclid algorithm[edit]

fn gcd(mut m: i32, mut n: i32) -> i32 {
   while m != 0 {
       let old_m = m;
       m = n % m;
       n = old_m;
   }
   n.abs()
}

Recursive Euclid algorithm[edit]

fn gcd(m: i32, n: i32) -> i32 {
   if m == 0 {
      n.abs()
   } else {
      gcd(n % m, m)
   }
}

Stein's Algorithm[edit]

Stein's algorithm is very much like Euclid's except that it uses bitwise operators (and consequently slightly more performant) and the integers must be unsigned. The following is a recursive implementation that leverages Rust's pattern matching.

use std::cmp::{min, max};
fn gcd(a: usize, b: usize) -> usize {
    match ((a, b), (a & 1, b & 1)) {
        ((x, y), _) if x == y               => y,
        ((0, x), _) | ((x, 0), _)           => x,
        ((x, y), (0, 1)) | ((y, x), (1, 0)) => gcd(x >> 1, y),
        ((x, y), (0, 0))                    => gcd(x >> 1, y >> 1) << 1,
        ((x, y), (1, 1))                    => { let (x, y) = (min(x, y), max(x, y)); 
                                                 gcd((y - x) >> 1, x) 
                                               }
        _                                   => unreachable!(),
    }
}

Tests[edit]

   println!("{}",gcd(399,-3999));
   println!("{}",gcd(0,3999));
   println!("{}",gcd(13*13,13*29));

3
3999
13

Sass/SCSS[edit]

Iterative Euclid's Algorithm

@function gcd($a,$b) {
	@while $b > 0 {
		$c: $a % $b;
		$a: $b;
		$b: $c;
	}
	@return $a;
}

Sather[edit]

Translation of: bc
class MATH is

  gcd_iter(u, v:INT):INT is
    loop while!( v.bool );
      t ::= u; u := v; v := t % v;
    end;
    return u.abs;
  end;

  gcd(u, v:INT):INT is
    if v.bool then return gcd(v, u%v); end;
    return u.abs;
  end;


  private swap(inout a, inout b:INT) is
    t ::= a;
    a := b;
    b := t;
  end;

  gcd_bin(u, v:INT):INT is
    t:INT;

    u := u.abs; v := v.abs;
    if u < v then swap(inout u, inout v); end;
    if v = 0 then return u; end;
    k ::= 1;
    loop while!( u.is_even and v.is_even );
      u := u / 2; v := v / 2;
      k := k * 2;
    end;
    if u.is_even then
      t := -v;
    else
      t := u;
    end;
    loop while!( t.bool );
      loop while!( t.is_even );
        t := t / 2;
      end;
      if t > 0 then 
        u := t;
      else
        v := -t;
      end;
      t := u - v;
    end;
    return u * k;
  end;

end;
class MAIN is
  main is
    a ::= 40902;
    b ::= 24140;
    #OUT + MATH::gcd_iter(a, b) + "\n";
    #OUT + MATH::gcd(a, b) + "\n";
    #OUT + MATH::gcd_bin(a, b) + "\n";
    -- built in
    #OUT + a.gcd(b) + "\n";
  end;
end;

Scala[edit]

def gcd(a: Int, b: Int): Int = if (b == 0) a.abs else gcd(b, a % b)

Using pattern matching

@tailrec
def gcd(a: Int, b: Int): Int = {
  b match {
    case 0 => a
    case _ => gcd(b, (a % b))
  }
}

Scheme[edit]

(define (gcd a b)
  (if (= b 0)
      a
      (gcd b (modulo a b))))

or using the standard function included with Scheme (takes any number of arguments):

(gcd a b)

Sed[edit]

#! /bin/sed -nf

# gcd.sed Copyright (c) 2010        by Paweł Zuzelski <pawelz@pld-linux.org>
# dc.sed  Copyright (c) 1995 - 1997 by Greg Ubben <gsu@romulus.ncsc.mil>

# usage:
#
#     echo N M | ./gcd.sed
#
# Computes the greatest common divisor of N and M integers using euclidean
# algorithm.

s/^/|P|K0|I10|O10|?~/

s/$/ [lalb%sclbsalcsblb0<F]sF sasblFxlap/

:next
s/|?./|?/
s/|?#[	 -}]*/|?/
/|?!*[lLsS;:<>=]\{0,1\}$/N
/|?!*[-+*/%^<>=]/b binop
/^|.*|?[dpPfQXZvxkiosStT;:]/b binop
/|?[_0-9A-F.]/b number
/|?\[/b string
/|?l/b load
/|?L/b Load
/|?[sS]/b save
/|?c/ s/[^|]*//
/|?d/ s/[^~]*~/&&/
/|?f/ s//&[pSbz0<aLb]dSaxsaLa/
/|?x/ s/\([^~]*~\)\(.*|?x\)~*/\2\1/
/|?[KIO]/ s/.*|\([KIO]\)\([^|]*\).*|?\1/\2~&/
/|?T/ s/\.*0*~/~/
#  a slow, non-stackable array implementation in dc, just for completeness
#  A fast, stackable, associative array implementation could be done in sed
#  (format: {key}value{key}value...), but would be longer, like load & save.
/|?;/ s/|?;\([^{}]\)/|?~[s}s{L{s}q]S}[S}l\1L}1-d0>}s\1L\1l{xS\1]dS{xL}/
/|?:/ s/|?:\([^{}]\)/|?~[s}L{s}L{s}L}s\1q]S}S}S{[L}1-d0>}S}l\1s\1L\1l{xS\1]dS{x/
/|?[ ~	cdfxKIOT]/b next
/|?\n/b next
/|?[pP]/b print
/|?k/ s/^\([0-9]\{1,3\}\)\([.~].*|K\)[^|]*/\2\1/
/|?i/ s/^\(-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}\)\(~.*|I\)[^|]*/\2\1/
/|?o/ s/^\(-\{0,1\}[1-9][0-9]*\.\{0,1\}[0-9]*\)\(~.*|O\)[^|]*/\2\1/
/|?[kio]/b pop
/|?t/b trunc
/|??/b input
/|?Q/b break
/|?q/b quit
h
/|?[XZz]/b count
/|?v/b sqrt
s/.*|?\([^Y]\).*/\1 is unimplemented/
s/\n/\\n/g
l
g
b next

:print
/^-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}~.*|?p/!b Print
/|O10|/b Print

#  Print a number in a non-decimal output base.  Uses registers a,b,c,d.
#  Handles fractional output bases (O<-1 or O>=1), unlike other dc's.
#  Converts the fraction correctly on negative output bases, unlike
#  UNIX dc.  Also scales the fraction more accurately than UNIX dc.
#
s,|?p,&KSa0kd[[-]Psa0la-]Sad0>a[0P]sad0=a[A*2+]saOtd0>a1-ZSd[[[[ ]P]sclb1\
!=cSbLdlbtZ[[[-]P0lb-sb]sclb0>c1+]sclb0!<c[0P1+dld>c]scdld>cscSdLbP]q]Sb\
[t[1P1-d0<c]scd0<c]ScO_1>bO1!<cO[16]<bOX0<b[[q]sc[dSbdA>c[A]sbdA=c[B]sbd\
B=c[C]sbdC=c[D]sbdD=c[E]sbdE=c[F]sb]xscLbP]~Sd[dtdZOZ+k1O/Tdsb[.5]*[.1]O\
X^*dZkdXK-1+ktsc0kdSb-[Lbdlb*lc+tdSbO*-lb0!=aldx]dsaxLbsb]sad1!>a[[.]POX\
+sb1[SbO*dtdldx-LbO*dZlb!<a]dsax]sadXd0<asbsasaLasbLbscLcsdLdsdLdLak[]pP,
b next

:Print
/|?p/s/[^~]*/&\
~&/
s/\(.*|P\)\([^|]*\)/\
\2\1/
s/\([^~]*\)\n\([^~]*\)\(.*|P\)/\1\3\2/
h
s/~.*//
/./{ s/.//; p; }
#  Just s/.//p would work if we knew we were running under the -n option.
#  Using l vs p would kind of do \ continuations, but would break strings.
g

:pop
s/[^~]*~//
b next

:load
s/\(.*|?.\)\(.\)/\20~\1/
s/^\(.\)0\(.*|r\1\([^~|]*\)~\)/\1\3\2/
s/.//
b next

:Load
s/\(.*|?.\)\(.\)/\2\1/
s/^\(.\)\(.*|r\1\)\([^~|]*~\)/|\3\2/
/^|/!i\
register empty
s/.//
b next

:save
s/\(.*|?.\)\(.\)/\2\1/
/^\(.\).*|r\1/ !s/\(.\).*|/&r\1|/
/|?S/ s/\(.\).*|r\1/&~/
s/\(.\)\([^~]*~\)\(.*|r\1\)[^~|]*~\{0,1\}/\3\2/
b next

:quit
t quit
s/|?[^~]*~[^~]*~/|?q/
t next
#  Really should be using the -n option to avoid printing a final newline.
s/.*|P\([^|]*\).*/\1/
q

:break
s/[0-9]*/&;987654321009;/
:break1
s/^\([^;]*\)\([1-9]\)\(0*\)\([^1]*\2\(.\)[^;]*\3\(9*\).*|?.\)[^~]*~/\1\5\6\4/
t break1
b pop

:input
N
s/|??\(.*\)\(\n.*\)/|?\2~\1/
b next

:count
/|?Z/ s/~.*//
/^-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}$/ s/[-.0]*\([^.]*\)\.*/\1/
/|?X/ s/-*[0-9A-F]*\.*\([0-9A-F]*\).*/\1/
s/|.*//
/~/ s/[^~]//g

s/./a/g
:count1
	s/a\{10\}/b/g
	s/b*a*/&a9876543210;/
	s/a.\{9\}\(.\).*;/\1/
	y/b/a/
/a/b count1
G
/|?z/ s/\n/&~/
s/\n[^~]*//
b next

:trunc
#  for efficiency, doesn't pad with 0s, so 10k 2 5/ returns just .40
#  The X* here and in a couple other places works around a SunOS 4.x sed bug.
s/\([^.~]*\.*\)\(.*|K\([^|]*\)\)/\3;9876543210009909:\1,\2/
:trunc1
	s/^\([^;]*\)\([1-9]\)\(0*\)\([^1]*\2\(.\)[^:]*X*\3\(9*\)[^,]*\),\([0-9]\)/\1\5\6\4\7,/
t trunc1
s/[^:]*:\([^,]*\)[^~]*/\1/
b normal

:number
s/\(.*|?\)\(_\{0,1\}[0-9A-F]*\.\{0,1\}[0-9A-F]*\)/\2~\1~/
s/^_/-/
/^[^A-F~]*~.*|I10|/b normal
/^[-0.]*~/b normal
s:\([^.~]*\)\.*\([^~]*\):[Ilb^lbk/,\1\2~0A1B2C3D4E5F1=11223344556677889900;.\2:
:digit
    s/^\([^,]*\),\(-*\)\([0-F]\)\([^;]*\(.\)\3[^1;]*\(1*\)\)/I*+\1\2\6\5~,\2\4/
t digit
s:...\([^/]*.\)\([^,]*\)[^.]*\(.*|?.\):\2\3KSb[99]k\1]SaSaXSbLalb0<aLakLbktLbk:
b next

:string
/|?[^]]*$/N
s/\(|?[^]]*\)\[\([^]]*\)]/\1|{\2|}/
/|?\[/b string
s/\(.*|?\)|{\(.*\)|}/\2~\1[/
s/|{/[/g
s/|}/]/g
b next

:binop
/^[^~|]*~[^|]/ !i\
stack empty
//!b next
/^-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}~/ !s/[^~]*\(.*|?!*[^!=<>]\)/0\1/
/^[^~]*~-\{0,1\}[0-9]*\.\{0,1\}[0-9]\{1,\}~/ !s/~[^~]*\(.*|?!*[^!=<>]\)/~0\1/
h
/|?\*/b mul
/|?\//b div
/|?%/b rem
/|?^/b exp

/|?[+-]/ s/^\(-*\)\([^~]*~\)\(-*\)\([^~]*~\).*|?\(-\{0,1\}\).*/\2\4s\3o\1\3\5/
s/\([^.~]*\)\([^~]*~[^.~]*\)\(.*\)/<\1,\2,\3|=-~.0,123456789<></
/^<\([^,]*,[^~]*\)\.*0*~\1\.*0*~/ s/</=/
:cmp1
	s/^\(<[^,]*\)\([0-9]\),\([^,]*\)\([0-9]\),/\1,\2\3,\4/
t cmp1
/^<\([^~]*\)\([^~]\)[^~]*~\1\(.\).*|=.*\3.*\2/ s/</>/
/|?/{
	s/^\([<>]\)\(-[^~]*~-.*\1\)\(.\)/\3\2/
	s/^\(.\)\(.*|?!*\)\1/\2!\1/
	s/|?![^!]\(.\)/&l\1x/
	s/[^~]*~[^~]*~\(.*|?\)!*.\(.*\)|=.*/\1\2/
	b next
}
s/\(-*\)\1|=.*/;9876543210;9876543210/
/o-/ s/;9876543210/;0123456789/
s/^>\([^~]*~\)\([^~]*~\)s\(-*\)\(-*o\3\(-*\)\)/>\2\1s\5\4/

s/,\([0-9]*\)\.*\([^,]*\),\([0-9]*\)\.*\([0-9]*\)/\1,\2\3.,\4;0/
:right1
	s/,\([0-9]\)\([^,]*\),;*\([0-9]\)\([0-9]*\);*0*/\1,\2\3,\4;0/
t right1
s/.\([^,]*\),~\(.*\);0~s\(-*\)o-*/\1~\30\2~/

:addsub1
	s/\(.\{0,1\}\)\(~[^,]*\)\([0-9]\)\(\.*\),\([^;]*\)\(;\([^;]*\(\3[^;]*\)\).*X*\1\(.*\)\)/\2,\4\5\9\8\7\6/
	s/,\([^~]*~\).\{10\}\(.\)[^;]\{0,9\}\([^;]\{0,1\}\)[^;]*/,\2\1\3/
#	  could be done in one s/// if we could have >9 back-refs...
/^~.*~;/!b addsub1

:endbin
s/.\([^,]*\),\([0-9.]*\).*/\1\2/
G
s/\n[^~]*~[^~]*//

:normal
s/^\(-*\)0*\([0-9.]*[0-9]\)[^~]*/\1\2/
s/^[^1-9~]*~/0~/
b next

:mul
s/\(-*\)\([0-9]*\)\.*\([0-9]*\)~\(-*\)\([0-9]*\)\.*\([0-9]*\).*|K\([^|]*\).*/\1\4\2\5.!\3\6,|\2<\3~\5>\6:\7;9876543210009909/

:mul1
    s/![0-9]\([^<]*\)<\([0-9]\{0,1\}\)\([^>]*\)>\([0-9]\{0,1\}\)/0!\1\2<\3\4>/
    /![0-9]/ s/\(:[^;]*\)\([1-9]\)\(0*\)\([^0]*\2\(.\).*X*\3\(9*\)\)/\1\5\6\4/
/<~[^>]*>:0*;/!t mul1

s/\(-*\)\1\([^>]*\).*/;\2^>:9876543210aaaaaaaaa/

:mul2
    s/\([0-9]~*\)^/^\1/
    s/<\([0-9]*\)\(.*[~^]\)\([0-9]*\)>/\1<\2>\3/

    :mul3
	s/>\([0-9]\)\(.*\1.\{9\}\(a*\)\)/\1>\2;9\38\37\36\35\34\33\32\31\30/
	s/\(;[^<]*\)\([0-9]\)<\([^;]*\).*\2[0-9]*\(.*\)/\4\1<\2\3/
	s/a[0-9]/a/g
	s/a\{10\}/b/g
	s/b\{10\}/c/g
    /|0*[1-9][^>]*>0*[1-9]/b mul3

    s/;/a9876543210;/
    s/a.\{9\}\(.\)[^;]*\([^,]*\)[0-9]\([.!]*\),/\2,\1\3/
    y/cb/ba/
/|<^/!b mul2
b endbin

:div
#  CDDET
/^[-.0]*[1-9]/ !i\
divide by 0
//!b pop
s/\(-*\)\([0-9]*\)\.*\([^~]*~-*\)\([0-9]*\)\.*\([^~]*\)/\2.\3\1;0\4.\5;0/
:div1
	s/^\.0\([^.]*\)\.;*\([0-9]\)\([0-9]*\);*0*/.\1\2.\3;0/
	s/^\([^.]*\)\([0-9]\)\.\([^;]*;\)0*\([0-9]*\)\([0-9]\)\./\1.\2\30\4.\5/
t div1
s/~\(-*\)\1\(-*\);0*\([^;]*[0-9]\)[^~]*/~123456789743222111~\2\3/
s/\(.\(.\)[^~]*\)[^9]*\2.\{8\}\(.\)[^~]*/\3~\1/
s,|?.,&SaSadSaKdlaZ+LaX-1+[sb1]Sbd1>bkLatsbLa[dSa2lbla*-*dLa!=a]dSaxsakLasbLb*t,
b next

:rem
s,|?%,&Sadla/LaKSa[999]k*Lak-,
b next

:exp
#  This decimal method is just a little faster than the binary method done
#  totally in dc:  1LaKLb [kdSb*LbK]Sb [[.5]*d0ktdSa<bkd*KLad1<a]Sa d1<a kk*
/^[^~]*\./i\
fraction in exponent ignored
s,[^-0-9].*,;9d**dd*8*d*d7dd**d*6d**d5d*d*4*d3d*2lbd**1lb*0,
:exp1
	s/\([0-9]\);\(.*\1\([d*]*\)[^l]*\([^*]*\)\(\**\)\)/;dd*d**d*\4\3\5\2/
t exp1
G
s,-*.\{9\}\([^9]*\)[^0]*0.\(.*|?.\),\2~saSaKdsaLb0kLbkK*+k1\1LaktsbkLax,
s,|?.,&SadSbdXSaZla-SbKLaLadSb[0Lb-d1lb-*d+K+0kkSb[1Lb/]q]Sa0>a[dk]sadK<a[Lb],
b next

:sqrt
#  first square root using sed:  8k2v at 1:30am Dec 17, 1996
/^-/i\
square root of negative number
/^[-0]/b next
s/~.*//
/^\./ s/0\([0-9]\)/\1/g
/^\./ !s/[0-9][0-9]/7/g
G
s/\n/~/
s,|?.,&K1+k KSbSb[dk]SadXdK<asadlb/lb+[.5]*[sbdlb/lb+[.5]*dlb>a]dsaxsasaLbsaLatLbk K1-kt,
b next

#  END OF GSU dc.sed

Seed7[edit]

const func integer: gcd (in var integer: a, in var integer: b) is func
  result
    var integer: gcd is 0;
  local
    var integer: help is 0;
  begin
    while a <> 0 do
      help := b rem a;
      b := a;
      a := help;
    end while;
    gcd := b;
  end func;

Original source: [1]

SequenceL[edit]

Tail Recursive Greatest Common Denominator using Euclidian Algorithm

gcd(a, b) :=
		a when b = 0
	else
		gcd(b, a mod b);

SETL[edit]

a := 33; b := 77;
print(" the gcd of",a," and ",b," is ",gcd(a,b));

c := 49865; d := 69811;
print(" the gcd of",c," and ",d," is ",gcd(c,d));

proc gcd (u, v);
  return if v = 0 then abs u else gcd (v, u mod v) end;
end;

Output:

the gcd of 33  and  77  is  11
the gcd of 49865  and  69811  is  9973

Sidef[edit]

Built-in[edit]

var arr = [100, 1_000, 10_000, 20];
say Math.gcd(arr...);

Recursive Euclid algorithm[edit]

func gcd(a, b) {
    b.is_zero ? a.abs : gcd(b, a % b);
}

Simula[edit]

For a recursive variant, see Sum multiples of 3 and 5.

BEGIN
    INTEGER PROCEDURE GCD(a, b); INTEGER a, b;
    BEGIN
        IF a = 0 THEN a := b
        ELSE
            WHILE 0 < b DO BEGIN INTEGER i;
                i := MOD(a, b); a := b; b := i;
            END;
        GCD := a
    END;

    INTEGER a, b;
    !outint(SYSOUT.IMAGE.MAIN.LENGTH, 0);!OUTIMAGE;!OUTIMAGE;
    !SYSOUT.IMAGE :- BLANKS(132);  ! this may or may not work;
    FOR b := 1 STEP 5 UNTIL 37 DO BEGIN
        FOR a := 0 STEP 2 UNTIL 21 DO BEGIN
            OUTTEXT("  ("); OUTINT(a, 0);
            OUTCHAR(','); OUTINT(b, 2);
            OUTCHAR(')'); OUTINT(GCD(a, b), 3);
        END;
        OUTIMAGE
    END
END
Output:
(0, 1)  1  (2, 1)  1  (4, 1)  1  (6, 1)  1  (8, 1)  1  (10, 1)  1  (12, 1)  1  (14, 1)  1  (16, 1)  1  (18, 1)  1  (20, 1)  1
(0, 6)  6  (2, 6)  2  (4, 6)  2  (6, 6)  6  (8, 6)  2  (10, 6)  2  (12, 6)  6  (14, 6)  2  (16, 6)  2  (18, 6)  6  (20, 6)  2
(0,11) 11  (2,11)  1  (4,11)  1  (6,11)  1  (8,11)  1  (10,11)  1  (12,11)  1  (14,11)  1  (16,11)  1  (18,11)  1  (20,11)  1
(0,16) 16  (2,16)  2  (4,16)  4  (6,16)  2  (8,16)  8  (10,16)  2  (12,16)  4  (14,16)  2  (16,16) 16  (18,16)  2  (20,16)  4
(0,21) 21  (2,21)  1  (4,21)  1  (6,21)  3  (8,21)  1  (10,21)  1  (12,21)  3  (14,21)  7  (16,21)  1  (18,21)  3  (20,21)  1
(0,26) 26  (2,26)  2  (4,26)  2  (6,26)  2  (8,26)  2  (10,26)  2  (12,26)  2  (14,26)  2  (16,26)  2  (18,26)  2  (20,26)  2
(0,31) 31  (2,31)  1  (4,31)  1  (6,31)  1  (8,31)  1  (10,31)  1  (12,31)  1  (14,31)  1  (16,31)  1  (18,31)  1  (20,31)  1
(0,36) 36  (2,36)  2  (4,36)  4  (6,36)  6  (8,36)  4  (10,36)  2  (12,36) 12  (14,36)  2  (16,36)  4  (18,36) 18  (20,36)  4

Slate[edit]

Slate's Integer type has gcd defined:

40902 gcd: 24140

Iterative Euclid algorithm[edit]

x@(Integer traits) gcd: y@(Integer traits)
"Euclid's algorithm for finding the greatest common divisor."
[| n m temp |
  n: x.
  m: y.
  [n isZero] whileFalse: [temp: n. n: m \\ temp. m: temp].
  m abs
].

Recursive Euclid algorithm[edit]

x@(Integer traits) gcd: y@(Integer traits)
[
  y isZero
    ifTrue: [x]
    ifFalse: [y gcd: x \\ y]
].

Smalltalk[edit]

The Integer class has its gcd method.

(40902 gcd: 24140) displayNl

An reimplementation of the Iterative Euclid's algorithm would be:

|gcd_iter|

gcd_iter := [ :a :b | 
  |u v| 
   u := a. v := b.
   [ v > 0 ]
     whileTrue: [ |t|
        t := u.
        u := v.
        v := t rem: v
     ].
   u abs
].

(gcd_iter value: 40902 value: 24140) printNl.

SNOBOL4[edit]

	define('gcd(i,j)')	:(gcd_end)
gcd	?eq(i,0)	:s(freturn)
	?eq(j,0)	:s(freturn)

loop	gcd = remdr(i,j)
	gcd = ?eq(gcd,0) j	:s(return)
	i = j
	j = gcd			:(loop)
gcd_end

	output = gcd(1071,1029)
end

Sparkling[edit]

function factors(n) {
	var f = {};

	for var i = 2; n > 1; i++ {
		while n % i == 0 {
			n /= i;
			f[i] = f[i] != nil ? f[i] + 1 : 1;
		}
	}

	return f;
}

function GCD(n, k) {
	let f1 = factors(n);
	let f2 = factors(k);

	let fs = map(f1, function(factor, multiplicity) {
		let m = f2[factor];
		return m == nil ? 0 : min(m, multiplicity);
	});

	let rfs = {};
	foreach(fs, function(k, v) {
		rfs[sizeof rfs] = pow(k, v);
	});

	return reduce(rfs, 1, function(x, y) { return x * y; });
}

function LCM(n, k) {
	return n * k / GCD(n, k);
}

SQL[edit]

Demonstration of Oracle 12c WITH Clause Enhancements

drop table tbl;
create table tbl
(
        u       number,
        v       number
);

insert into tbl ( u, v ) values ( 20, 50 );
insert into tbl ( u, v ) values ( 21, 50 );
insert into tbl ( u, v ) values ( 21, 51 );
insert into tbl ( u, v ) values ( 22, 50 );
insert into tbl ( u, v ) values ( 22, 55 );

commit;

with
        function gcd ( ui in number, vi in number )
        return number
        is
                u number := ui;
                v number := vi;
                t number;
        begin
                while v > 0
                loop
                        t := u;
                        u := v;
                        v:= mod(t, v );
                end loop;
                return abs(u);
        end gcd;
        select u, v, gcd ( u, v )
        from tbl
/
Output:
Table dropped.


Table created.


1 row created.


1 row created.


1 row created.


1 row created.


1 row created.


Commit complete.


         U          V   GCD(U,V)
---------- ---------- ----------
        20         50         10
        21         50          1
        21         51          3
        22         50          2
        22         55         11

Demonstration of SQL Server 2008

CREATE FUNCTION gcd (
  @ui INT,
  @vi INT
) RETURNS INT

AS

BEGIN
    DECLARE @t INT
    DECLARE @u INT
    DECLARE @v INT
	
    SET @u = @ui
    SET @v = @vi

    WHILE @v > 0
    BEGIN
        SET @t = @u;
        SET @u = @v;
        SET @v = @t % @v;
    END;
    RETURN abs( @u );
END

GO

CREATE TABLE tbl (
  u INT,
  v INT
);
 
INSERT INTO tbl ( u, v ) VALUES ( 20, 50 );
INSERT INTO tbl ( u, v ) VALUES ( 21, 50 );
INSERT INTO tbl ( u, v ) VALUES ( 21, 51 );
INSERT INTO tbl ( u, v ) VALUES ( 22, 50 );
INSERT INTO tbl ( u, v ) VALUES ( 22, 55 );

SELECT u, v, dbo.gcd ( u, v )
  FROM tbl;

DROP TABLE tbl;

DROP FUNCTION gcd;

PostgreSQL function using a recursive common table expression

CREATE FUNCTION gcd(integer, integer)
RETURNS integer
LANGUAGE sql
AS $function$
WITH RECURSIVE x (u, v) AS (
  SELECT ABS($1), ABS($2)
  UNION
  SELECT v, u % v FROM x WHERE v > 0
)
SELECT min(u) FROM x;
$function$
Output:
postgres> select gcd(40902, 24140);
gcd
-----
34
SELECT 1
Time: 0.012s

Standard ML[edit]

See also #ML / Standard ML.

(* Euclid’s algorithm. *)

fun gcd (u, v) =
    let
        fun loop (u, v) =
            if v = 0 then
                u
            else
                loop (v, u mod v)
    in
        loop (abs u, abs v)
    end

(* Using the Rosetta Code example for assertions in Standard ML. *)
fun assert cond =
    if cond then () else raise Fail "assert"
 
val () = assert (gcd (0, 0) = 0)
val () = assert (gcd (0, 10) = 10)
val () = assert (gcd (~10, 0) = 10)
val () = assert (gcd (9, 6) = 3)
val () = assert (gcd (~6, ~9) = 3)
val () = assert (gcd (40902, 24140) = 34)
val () = assert (gcd (40902, ~24140) = 34)
val () = assert (gcd (~40902, 24140) = 34)
val () = assert (gcd (~40902, ~24140) = 34)
val () = assert (gcd (24140, 40902) = 34)
val () = assert (gcd (~24140, 40902) = 34)
val () = assert (gcd (24140, ~40902) = 34)
val () = assert (gcd (~24140, ~40902) = 34)

Stata[edit]

function gcd(a_,b_) {
	a = abs(a_)
	b = abs(b_)
	while (b>0) {
		a = mod(a,b)
		swap(a,b)
	}
	return(a)
}

Swift[edit]

// Iterative

func gcd(var a: Int, var b: Int) -> Int {
    
    a = abs(a); b = abs(b)
    
    if (b > a) { swap(&a, &b) }

    while (b > 0) { (a, b) = (b, a % b) }
    
    return a
}

// Recursive

func gcdr (var a: Int, var b: Int) -> Int {
    
    a = abs(a); b = abs(b)

    if (b > a) { swap(&a, &b) }
    
    return gcd_rec(a,b)
}


private func gcd_rec(a: Int, b: Int) -> Int {
    
    return b == 0 ? a : gcd_rec(b, a % b)
}


for (a,b) in [(1,1), (100, -10), (10, -100), (-36, -17), (27, 18), (30, -42)] {
    
    println("Iterative: GCD of \(a) and \(b) is \(gcd(a, b))")
    println("Recursive: GCD of \(a) and \(b) is \(gcdr(a, b))")
}
Output:
Iterative: GCD of 1 and 1 is 1
Recursive: GCD of 1 and 1 is 1
Iterative: GCD of 100 and -10 is 10
Recursive: GCD of 100 and -10 is 10
Iterative: GCD of 10 and -100 is 10
Recursive: GCD of 10 and -100 is 10
Iterative: GCD of -36 and -17 is 1
Recursive: GCD of -36 and -17 is 1
Iterative: GCD of 27 and 18 is 9
Recursive: GCD of 27 and 18 is 9
Iterative: GCD of 30 and -42 is 6
Recursive: GCD of 30 and -42 is 6

Tcl[edit]

Iterative Euclid algorithm[edit]

package require Tcl 8.5
namespace path {::tcl::mathop ::tcl::mathfunc}
proc gcd_iter {p q} {
    while {$q != 0} {
        lassign [list $q [% $p $q]] p q
    }
    abs $p
}

Recursive Euclid algorithm[edit]

proc gcd {p q} {
    if {$q == 0} {
        return $p
    }
    gcd $q [expr {$p % $q}]
}

With Tcl 8.6, this can be optimized slightly to:

proc gcd {p q} {
    if {$q == 0} {
        return $p
    }
    tailcall gcd $q [expr {$p % $q}]
}

(Tcl does not perform automatic tail-call optimization introduction because that makes any potential error traces less informative.)

Iterative binary algorithm[edit]

package require Tcl 8.5
namespace path {::tcl::mathop ::tcl::mathfunc}
proc gcd_bin {p q} {
    if {$p == $q} {return [abs $p]}
    set p [abs $p]
    if {$q == 0} {return $p}
    set q [abs $q]
    if {$p < $q} {lassign [list $q $p] p q}
    set k 1
    while {($p & 1) == 0 && ($q & 1) == 0} {
        set p [>> $p 1]
        set q [>> $q 1]
        set k [<< $k 1]
    }
    set t [expr {$p & 1 ? -$q : $p}]
    while {$t} {
        while {$t & 1 == 0} {set t [>> $t 1]}
        if {$t > 0} {set p $t} {set q [- $t]}
        set t [- $p $q]
    }
    return [* $p $k]
}

Notes on performance[edit]

foreach proc {gcd_iter gcd gcd_bin} {
    puts [format "%-8s - %s" $proc [time {$proc $u $v} 100000]]
}

Outputs:

gcd_iter - 4.46712 microseconds per iteration
gcd      - 5.73969 microseconds per iteration
gcd_bin  - 9.25613 microseconds per iteration

Transact-SQL[edit]

CREATE OR ALTER FUNCTION [dbo].[PGCD]
    (    @a BigInt
    ,    @b BigInt
    )
RETURNS BigInt
WITH RETURNS NULL ON NULL INPUT
-- Calculates the Greatest Common Denominator of two numbers (1 if they are coprime).
BEGIN
DECLARE @PGCD BigInt;

WITH    Vars(A, B)
As  (   SELECT  Max(V.N) As A
            ,   Min(V.N) As B
        FROM (  VALUES  ( Abs(@a) , Abs(@b)) ) Params(A, B)
        -- First, get absolute value
        Cross APPLY (   VALUES (Params.A) , (Params.B) ) V(N)
        -- Then, order parameters without Greatest/Least functions
        WHERE Params.A > 0
            And Params.B > 0 -- If 0 passed in, NULL shall be the output
    )
    ,   Calc(A, B)
As  (   SELECT  A
            ,   B
        FROM    Vars

        UNION ALL

        SELECT  B As A
            ,   A % B As B -- Self-ordering
        FROM    Calc
        WHERE   Calc.A > 0
            And Calc.B > 0
    )
SELECT  @PGCD = Min(A)
FROM    Calc
WHERE   Calc.B = 0
;

RETURN @PGCD;

END

TSE SAL[edit]

// library: math: get: greatest: common: divisor <description>greatest common divisor whole numbers. Euclid's algorithm. Recursive version</description> <version control></version control> <version>1.0.0.0.3</version> <version control></version control> (filenamemacro=getmacdi.s) [<Program>] [<Research>] [kn, ri, su, 20-01-2013 14:22:41]
INTEGER PROC FNMathGetGreatestCommonDivisorI( INTEGER x1I, INTEGER x2I )
 //
 IF ( x2I == 0 )
  //
  RETURN( x1I )
  //
 ENDIF
 //
 RETURN( FNMathGetGreatestCommonDivisorI( x2I, x1I MOD x2I ) )
 //
END

PROC Main()
 STRING s1[255] = "353"
 STRING s2[255] = "46"
 REPEAT
  IF ( NOT ( Ask( " = ", s1, _EDIT_HISTORY_ ) ) AND ( Length( s1 ) > 0 ) ) RETURN() ENDIF
  IF ( NOT ( Ask( " = ", s2, _EDIT_HISTORY_ ) ) AND ( Length( s2 ) > 0 ) ) RETURN() ENDIF
  Warn( FNMathGetGreatestCommonDivisorI( Val( s1 ), Val( s2 ) ) ) // gives e.g. 1
 UNTIL FALSE
END

TXR[edit]

$ txr -p '(gcd (expt 2 123) (expt 6 49))'
562949953421312

TypeScript[edit]

Iterative implementation

function gcd(a: number, b: number) {
  a = Math.abs(a);
  b = Math.abs(b);

  if (b > a) {
    let temp = a;
    a = b;
    b = temp; 
  }

  while (true) {
    a %= b;
    if (a === 0) { return b; }
    b %= a;
    if (b === 0) { return a; }
  }
}

Recursive.

function gcd_rec(a: number, b: number) {
  return b ? gcd_rec(b, a % b) : Math.abs(a);
}

UNIX Shell[edit]

Works with: Bourne Shell
gcd() {
	# Calculate $1 % $2 until $2 becomes zero.
	until test 0 -eq "$2"; do
		# Parallel assignment: set -- 1 2
		set -- "$2" "`expr "$1" % "$2"`"
	done

	# Echo absolute value of $1.
	test 0 -gt "$1" && set -- "`expr 0 - "$1"`"
	echo "$1"
}

gcd -47376 87843
# => 987

dash or bash[edit]

Procedural :

gcd() { until test 0 -eq "$2";do set -- "$2" "$(($1 % $2))";done;if [ 0 -gt "$1" ];then echo "$((- $1))";else  echo "$1"; fi }

gcd -47376 87843
# => 987


Recursive :

gcd () { if [ "$2" -ne 0 ];then gcd "$2" "$(($1 % $2))";else echo "$1";fi }

gcd 100 75
# => 25

C Shell[edit]

alias gcd eval \''set gcd_args=( \!*:q )	\\
	@ gcd_u=$gcd_args[2]			\\
	@ gcd_v=$gcd_args[3]			\\
	while ( $gcd_v != 0 )			\\
		@ gcd_t = $gcd_u % $gcd_v	\\
		@ gcd_u = $gcd_v		\\
		@ gcd_v = $gcd_t		\\
	end					\\
	if ( $gcd_u < 0 ) @ gcd_u = - $gcd_u	\\
	@ $gcd_args[1]=$gcd_u			\\
'\'

gcd result -47376 87843
echo $result
# => 987

Ursa[edit]

import "math"
out (gcd 40902 24140) endl console
Output:
34

Ursala[edit]

This doesn't need to be defined because it's a library function, but it can be defined like this based on a recursive implementation of Euclid's algorithm. This isn't the simplest possible solution because it includes a bit shifting optimization that happens when both operands are even.

#import nat

gcd = ~&B?\~&Y ~&alh^?\~&arh2faltPrXPRNfabt2RCQ @a ~&ar^?\~&al ^|R/~& ^/~&r remainder

test program:

#cast %nWnAL

test = ^(~&,gcd)* <(25,15),(36,16),(120,45),(30,100)>

output:

<
   (25,15): 5,
   (36,16): 4,
   (120,45): 15,
   (30,100): 10>

V[edit]

like joy

iterative[edit]

[gcd
   [0 >] [dup rollup %]
   while
   pop
].

recursive[edit]

like python

[gcd
   [zero?] [pop]
      [swap [dup] dip swap %]
   tailrec].

same with view: (swap [dup] dip swap % is replaced with a destructuring view)

[gcd
   [zero?] [pop]
     [[a b : [b a b %]] view i]
   tailrec].

running it

|1071 1029 gcd
=21

Verilog[edit]

module gcd
  (
  input reset_l,
  input clk,

  input [31:0] initial_u,
  input [31:0] initial_v,
  input load,

  output reg [31:0] result,
  output reg busy
  );

reg [31:0] u, v;

always @(posedge clk or negedge reset_l)
  if (!reset_l)
    begin
      busy <= 0;
      u <= 0;
      v <= 0;
    end
  else
    begin

      result <= u + v; // Result (one of them will be zero)

      busy <= u && v; // We're still busy...

      // Repeatedly subtract smaller number from larger one
      if (v <= u)
        u <= u - v;
      else if (u < v)
        v <= v - u;

      if (load) // Load new problem when high
        begin
          u <= initial_u;
          v <= initial_v;
          busy <= 1;
        end

    end

endmodule

V (Vlang)[edit]

Iterative[edit]

fn gcd(xx int, yy int) int {
    mut x, mut y := xx, yy
    for y != 0 {
        x, y = y, x%y
    }
    return x
}
 
fn main() {
    println(gcd(33, 77))
    println(gcd(49865, 69811))
}

Builtin[edit]

(This is just a wrapper for big.gcd)

import math.big
fn gcd(x i64, y i64) i64 {
    return big.integer_from_i64(x).gcd(big.integer_from_i64(y)).int()
}
 
fn main() {
    println(gcd(33, 77))
    println(gcd(49865, 69811))
}
Output in either case:
11
9973

Wortel[edit]

Operator

@gcd a b

Number expression

!#~kg a b

Iterative

&[a b] [@vars[t] @while b @:{t b b %a b a t} a]

Recursive

&{gcd a b} ?{b !!gcd b %a b @abs a}

Wren[edit]

var gcd = Fn.new { |x, y|
    while (y != 0) {
        var t = y
        y = x % y
        x = t
    }
    return x.abs
}

System.print("gcd(33, 77) = %(gcd.call(33, 77))")
System.print("gcd(49865, 69811) = %(gcd.call(49865, 69811))")
Output:
gcd(33, 77) = 11
gcd(49865, 69811) = 9973

x86 Assembly[edit]

Using GNU Assembler syntax:

.text
.global pgcd

pgcd:
        push    %ebp
        mov     %esp, %ebp

        mov     8(%ebp), %eax
        mov     12(%ebp), %ecx
        push    %edx

.loop:
        cmp     $0, %ecx
        je      .end
        xor     %edx, %edx
        div     %ecx
        mov     %ecx, %eax
        mov     %edx, %ecx
        jmp     .loop

.end:
        pop     %edx
        leave
        ret

XLISP[edit]

GCD is a built-in function. If we wanted to reimplement it, one (tail-recursive) way would be like this:

(defun greatest-common-divisor (x y)
	(if (= y 0)
		x
		(greatest-common-divisor y (mod x y)) ) )

XPL0[edit]

include c:\cxpl\codes;

func GCD(U, V); \Return the greatest common divisor of U and V
int  U, V;
int  T;
[while V do     \Euclid's method
    [T:= U;  U:= V;  V:= rem(T/V)];
return abs(U);
];

\Display the GCD of two integers entered on command line
IntOut(0, GCD(IntIn(8), IntIn(8)))

Z80 Assembly[edit]

Uses the iterative subtraction implementation of Euclid's algorithm because the Z80 does not implement modulus or division opcodes.

; Inputs: a, b
; Outputs: a = gcd(a, b)
; Destroys: c
; Assumes: a and b are positive one-byte integers
gcd:
    cp b
    ret z                   ; while a != b

    jr c, else              ; if a > b

    sub b                   ; a = a - b

    jr gcd

else:
    ld c, a                 ; Save a
    ld a, b                 ; Swap b into a so we can do the subtraction
    sub c                   ; b = b - a
    ld b, a                 ; Put a and b back where they belong
    ld a, c

    jr gcd

zkl[edit]

This is a method on integers:

(123456789).gcd(987654321) //-->9

Using the gnu big num library (GMP):

var BN=Import("zklBigNum");
BN(123456789).gcd(987654321) //-->9

or

fcn gcd(a,b){ while(b){ t:=a; a=b; b=t%b } a.abs() }