Smallest multiple

Task description is taken from Project Euler
(https://projecteuler.net/problem=5)
2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

Smallest multiple is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Related

11l

```F f(n)
V ans = BigInt(1)
L(i) 1..n
ans *= BigInt(i) I/ gcd(BigInt(i), BigInt(ans))
R ans

L(n) [10, 20, 200, 2000]
print(n‘: ’f(n))```
Output:
```10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000
```

ALGOL 68

Translation of: Wren
Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses Algol 68G's LONG LONG INT which has specifiable precision.

```BEGIN # find the smallest number that is divisible by each of the numbers 1..n #
# translation of the Wren sample #
PR precision 1000 PR # set the precision of LONG LONG INT #
# returns the lowest common multiple of the numbers 1 : n #
PROC lcm = ( INT n )LONG LONG INT:
BEGIN
# sieve the primes to n #
[]BOOL prime = PRIMESIEVE n;
LONG LONG INT result := 1;
FOR p TO UPB prime DO
IF prime[ p ] THEN
LONG LONG INT f := p;           # f will be set to the #
WHILE f * p <= n DO f *:= p OD; # highest multiple of p <= n #
result *:= f
FI
OD;
result
END # lcm # ;
# returns a string representation of n with commas #
PROC commatise = ( LONG LONG INT n )STRING:
BEGIN
STRING result      := "";
STRING unformatted  = whole( n, 0 );
INT    ch count    := 0;
FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
IF   ch count <= 2 THEN ch count +:= 1
ELSE                    ch count  := 1; "," +=: result
FI;
unformatted[ c ] +=: result
OD;
result
END; # commatise #
print( ( "The LCMs of the numbers 1 to N inclusive is:", newline ) );
[]INT tests = ( 10, 20, 200, 2000 );
FOR i FROM LWB tests TO UPB tests DO
print( ( whole( tests[ i ], -5 ), ": ", commatise( lcm( tests[ i ] ) ), newline ) )
OD
END```
Output:
```   10: 2,520
20: 232,792,560
200: 337,293,588,832,926,264,639,465,766,794,841,407,432,394,382,785,157,234,228,847,021,917,234,018,060,677,390,066,992,000
2000: 151,117,794,877,444,315,307,536,308,337,572,822,173,736,308,853,579,339,903,227,904,473,000,476,322,347,234,655,122,160,866,668,946,941,993,951,014,270,933,512,030,194,957,221,371,956,828,843,521,568,082,173,786,251,242,333,157,830,450,435,623,211,664,308,500,316,844,478,617,809,101,158,220,672,108,895,053,508,829,266,120,497,031,742,749,376,045,929,890,296,052,805,527,212,315,382,805,219,353,316,270,742,572,401,962,035,464,878,235,703,759,464,796,806,075,131,056,520,079,836,955,770,415,021,318,508,272,982,103,736,658,633,390,411,347,759,000,563,271,226,062,182,345,964,184,167,346,918,225,243,856,348,794,013,355,418,404,695,826,256,911,622,054,015,423,611,375,261,945,905,974,225,257,659,010,379,414,787,547,681,984,112,941,581,325,198,396,634,685,659,217,861,208,771,400,322,507,388,161,967,513,719,166,366,839,894,214,040,787,733,471,287,845,629,833,993,885,413,462,225,294,548,785,581,641,804,620,417,256,563,685,280,586,511,301,918,399,010,451,347,815,776,570,842,790,738,545,306,707,750,937,624,267,501,103,840,324,470,083,425,714,138,183,905,657,667,736,579,430,274,197,734,179,172,691,637,931,540,695,631,396,056,193,786,415,805,463,680,000
```

Asymptote

```int temp = 2*3*5*7*11*13*17*19;
int smalmul = temp;
int lim = 1;
while (lim <= 20) {
lim = lim + 1;
while (smalmul % lim != 0) {
lim = 1;
smalmul = smalmul + temp;
}
}
write(smalmul);
```

AutoHotkey

```primes := 1
loop 20
if prime_numbers(A_Index).Count() = 1
primes *= A_Index

loop
{
Result := A_Index*primes
loop 20
if Mod(Result, A_Index)
continue, 2
break
}
MsgBox % Result
return

prime_numbers(n) { ; http://www.rosettacode.org/wiki/Prime_decomposition#Optimized_Version
if (n <= 3)
return [n]
ans := [], done := false
while !done
{
if !Mod(n,2){
ans.push(2), n /= 2
continue
}
if !Mod(n,3) {
ans.push(3), n /= 3
continue
}
if (n = 1)
return ans
sr := sqrt(n), done := true
; try to divide the checked number by all numbers till its square root.
i := 6
while (i <= sr+6){
if !Mod(n, i-1) { ; is n divisible by i-1?
ans.push(i-1), n /= i-1, done := false
break
}
if !Mod(n, i+1) { ; is n divisible by i+1?
ans.push(i+1), n /= i+1, done := false
break
}
i += 6
}
}
ans.push(n)
return ans
}
```
Output:
`232792560`

BASIC

BASIC256

```temp = 2*3*5*7*11*13*17*19
smalmul = temp
lim = 1
do
lim += 1
if (smalmul mod lim) then lim = 1 : smalmul += temp
until lim = 20
print smalmul
```
Output:
`232792560`

PureBasic

```OpenConsole()
temp.i = 2*3*5*7*11*13*17*19
smalmul.i = temp
lim.i = 1
Repeat
lim + 1
If (smalmul % lim)
lim = 1
smalmul = smalmul + temp
EndIf
Until lim = 20
PrintN(Str(smalmul))
Input()
CloseConsole()
```
Output:
`232792560`

True BASIC

```LET temp = 2*3*5*7*11*13*17*19
LET smalmul = temp
LET lim = 1
DO
LET lim = lim+1
IF (REMAINDER(ROUND(smalmul),ROUND(lim)) <> 0) THEN
LET lim = 1
LET smalmul = smalmul+temp
END IF
LOOP UNTIL lim = 20
PRINT smalmul
END
```
Output:
`232792560`

F#

This task uses Extensible Prime Generator (F#)

```// Least Multiple. Nigel Galloway: October 22nd., 2021
let fG n g=let rec fN i=match i*g with g when n>g->fN g |_->i in fN g
let leastMult n=let fG=fG n in primes32()|>Seq.takeWhile((>=)n)|>Seq.map fG|>Seq.reduce((*))
printfn \$"%d{leastMult 20}"
```
Output:
```232792560
```

Factor

Works with: Factor version 0.98
```USING: math.functions math.ranges prettyprint sequences ;

20 [1,b] 1 [ lcm ] reduce .
```
Output:
```232792560
```

Fermat

```Func Ilog( n, b ) =
i:=0;                  {integer logarithm of n to base b, positive only}
while b^i<=n do
i:+;
od;
i-1.;

Func Smalmul( n ) =
s:=1;
for a = 1 to n do
if Isprime(a) then s:=s*a^Ilog(n, a) fi;
od;
s.;

!Smalmul(20);```
Output:
`232792560`

FreeBASIC

Use the code from the Least common multiple example as an include.

```#include"lcm.bas"

redim shared as ulongint smalls(0 to 1)  'calculate and store as we go
smalls(0) = 0: smalls(1) = 1

function smalmul(n as longint) as ulongint
if n<0 then return smalmul(-n)     'deal with negative input
dim as uinteger m = ubound(smalls)
if n<=m then return smalls(n)  'have we calculated this already
'if not, make room for the next bunch of terms
redim preserve as ulongint smalls(0 to n)
for i as uinteger = m+1 to n
smalls(i) = lcm(smalls(i-1), i)
next i
return smalls(n)
end function

for i as uinteger = 0 to 20
print i, smalmul(i)
next i
```

Go

Translation of: Wren
Library: Go-rcu
```package main

import (
"fmt"
"math/big"
"rcu"
)

func lcm(n int) *big.Int {
lcm := big.NewInt(1)
t := new(big.Int)
for _, p := range rcu.Primes(n) {
f := p
for f*p <= n {
f *= p
}
lcm.Mul(lcm, t.SetUint64(uint64(f)))
}
return lcm
}

func main() {
fmt.Println("The LCMs of the numbers 1 to N inclusive is:")
for _, i := range []int{10, 20, 200, 2000} {
fmt.Printf("%4d: %s\n", i, lcm(i))
}
}
```
Output:
```The LCMs of the numbers 1 to N inclusive is:
10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000
```

```import Text.Printf (printf)

--- SMALLEST INTEGER EVENLY DIVISIBLE BY EACH OF [1..N] --

smallest :: Integer -> Integer
smallest =
foldr lcm 1 . enumFromTo 1

--------------------------- TEST -------------------------
main :: IO ()
main =
(putStrLn . unlines) \$
showSmallest <\$> [10, 20, 200, 2000]

------------------------- DISPLAY ------------------------
showSmallest :: Integer -> String
showSmallest =
((<>) . (<> " -> ") . printf "%4d")
<*> (printf "%d" . smallest)
```
Output:
```  10 -> 2520
20 -> 232792560
200 -> 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000 -> 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000```

jq

Works with jq (*)
Works with gojq, the Go implementation of jq

The following uses `is_prime` as defined at Erdős-primes#jq.

(*) The C implementation of jq has sufficient accuracy for N == 20 but not N == 200, so the output shown below is based on a run of gojq.

```# Output: a stream of primes less than \$n in increasing order
def primes(\$n):
2, (range(3; \$n; 2) | select(is_prime));

# lcm of 1 to \$n inclusive
def lcm:
. as \$n
| reduce primes(\$n) as \$p (1;
. * (\$p | until(. * \$p > \$n; . * \$p)) ) ;

"N: LCM of the numbers 1 to N inclusive",
( 10, 20, 200, 2000
| "\(.): \(smallest_multiple)" )```
Output:
```N: LCM of the numbers 1 to N inclusive
10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000
```

Julia

```julia> foreach(x -> @show(lcm(x)), [1:10, 1:20, big"1":200, big"1":2000])
lcm(x) = 2520
lcm(x) = 232792560
lcm(x) = 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
lcm(x) = 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000
```

Mathematica / Wolfram Language

```LCM @@ Range[20]
```
Output:
```
232792560

```

Pascal

Here the simplest way, like Raku, check the highest exponent of every prime in range
Using harded coded primes.

```{\$IFDEF FPC}
{\$MODE DELPHI}
{\$ELSE}
{\$APPTAYPE CONSOLE}
{\$ENDIF}
const
smallprimes : array[0..10] of Uint32 = (2,3,5,7,11,13,17,19,23,29,31);
MAX = 20;

function getmaxfac(pr: Uint32): Uint32;
//get the pr^highest exponent of prime used in 2 .. MAX
var
i,fac : integer;
Begin
result := pr;
while pr*result <= MAX do
result *= pr;
end;

var
n,pr,prIdx : Uint32;
BEGIN
n := 1;
prIdx := 0;
pr := smallprimes[prIdx];
repeat
pr := smallprimes[prIdx];
n *= getmaxfac(pr);
inc(prIdx);
pr := smallprimes[prIdx];
until pr>MAX;
writeln(n);
{\$IFDEF WINDOWS}
{\$ENDIF}
END.
```
Output:
```  232792560
```

extended

fascinating find, that the count of digits is nearly a constant x upper rangelimit.
The number of factors is the count of primes til limit.See GetFactorList.
No need for calculating lcm(lcm(lcm(1,2),3),4..) or prime decomposition
Using prime sieve.

```{\$IFDEF FPC}
{\$MODE DELPHI} {\$Optimization On}
{\$ELSE}
{\$APPTAYPE CONSOLE}
{\$ENDIF}
{\$DEFINE USE_GMP}
uses
{\$IFDEF USE_GMP}
gmp,
{\$ENDIF}
sysutils; //format
const
MAX_LIMIT = 2*1000*1000;
UpperLimit = MAX_LIMIT+1000;// so to find a prime beyond MAX_LIMIT
MAX_UINT64 = 46;// unused.Limit to get an Uint64 output
type
tFactors = array of Uint32;
tprimelist = array of byte;
var
primeDeltalist : tPrimelist;
factors,
saveFactors:tFactors;
saveFactorsIdx,
maxFactorsIdx : Uint32;
procedure Init_Primes;
var
pPrime : pByte;
p,i,delta,cnt: NativeUInt;
begin
setlength(primeDeltalist,UpperLimit+3*8+1);
pPrime := @primeDeltalist[0];
//delete multiples of 2,3
i := 0;
repeat
//take care of endianess //0706050403020100
pUint64(@pPrime[i+0])^ := \$0100010000000100;
pUint64(@pPrime[i+8])^ := \$0000010001000000;
pUint64(@pPrime[i+16])^:= \$0100000001000100;
inc(i,24);
until i>UpperLimit;
cnt := 2;// 2,3
p := 5;
delta := 1;//5-3
repeat
if pPrime[p] <> 0 then
begin
i := p*p;
if i > UpperLimit then
break;
inc(cnt);
pPrime[p-2*delta] := delta;
delta := 0;
repeat
pPrime[i] := 0;
inc(i,2*p);
until i>UpperLimit;
end;
inc(p,2);
inc(delta);
until p*p>UpperLimit;
setlength(saveFactors,cnt);
//convert to delta
repeat
if pPrime[p]<> 0 then
begin
pPrime[p-2*delta] := delta;
inc(cnt);
delta := 0;
end;
inc(p,2);
inc(delta);
until p > UpperLimit;
setlength(factors,cnt);
factors[0] := 2;
factors[1] := 3;
i := 2;
p := 5;
repeat
factors[i] := p;
p += 2*pPrime[p];
i += 1;
until i >= cnt;
setlength(primeDeltalist,0);
//  writeln(length(savefactors)); writeln(length(factors));
end;

{\$IFDEF USE_GMP}
procedure ConvertToMPZ(const factors:tFactors;dgtCnt:UInt32);
const
c19Digits = QWord(10*1000000)*1000000*1000000;
var
mp,mpdiv : mpz_t;
s : AnsiString;
rest,last : Uint64;
f : Uint32;
i :int32;
begin
//Init and allocate space
mpz_init_set_ui(mp,0);
mpz_init(mpdiv);
mpz_ui_pow_ui(mpdiv,10,dgtCnt);
mpz_set_ui(mp,1);

i := maxFactorsIdx;
rest := 1;
repeat
last := rest;
f := factors[i];
rest *= f;
if rest div f <> last then
begin
mpz_mul_ui(mp,mp,last);
rest := f;
end;
dec(i);
until i < 0;
mpz_mul_ui(mp,mp,rest);

If dgtcnt>40 then
begin
rest := mpz_fdiv_ui(mp,c19Digits);
s := '..'+Format('%.19u',[rest]);
mpz_fdiv_q_ui (mpdiv,mpdiv,c19Digits);
mpz_fdiv_q(mp,mp,mpdiv);
rest := mpz_get_ui(mp);
writeln(rest:19,s);
mpz_clear(mpdiv);
end
else
Begin
setlength(s,dgtCnt+1000);
mpz_get_str(@s[1],10,mp);
writeln(s);
i := length(s);
while not(s[i] in['0'..'9']) do
dec(i);
setlength(s,i+1);
writeln(s);
end;
mpz_clear(mp);
end;
{\$ENDIF}

procedure CheckDigits(const factors:tFactors);
var
dgtcnt : extended;
i : integer;
begin
dgtcnt := 0;
i := 0;
repeat
dgtcnt += ln(factors[i]);
inc(i);
until i > maxFactorsIdx;
dgtcnt := trunc(dgtcnt/ln(10))+1;
writeln(' has ',maxFactorsIdx+1:10,' factors and ',dgtcnt:10:0,' digits');
{\$IFDEF USE_GMP}
i := trunc(dgtcnt);
if i < 1000*1000 then
ConvertToMPZ(factors,i);
{\$ENDIF}
end;

function ConvertToUint64(const factors:tFactors):Uint64;
var
i : integer;
begin
if maxFactorsIdx >15 then
Exit(0);
result := 1;
for i := 0 to maxFactorsIdx do
result *= factors[i];
end;

function ConvertToStr(const factors:tFactors):Ansistring;
var
s : Ansistring;
i : integer;
begin
result := '';
for i := 0 to maxFactorsIdx-1 do
begin
str(factors[i],s);
result += s+'*';
end;
str(factors[maxFactorsIdx],s);
result += s;
end;

procedure GetFactorList(var factors:tFactors;max:Uint32);
var
p,f,lf : Uint32;
BEGIN
p := 2;
lf := 0;
saveFactors[lf] := p;
while p*p <= max do
Begin
saveFactors[lf] := p;
f := p*p;
while f*p <= max do
f*= p;
factors[lf] := f;
inc(lf);
p := factors[lf];
if p= 0 then HALT;
end;
if lf>0 then
saveFactorsIdx := lf-1;
repeat
inc(lf)
until factors[lf]>Max;
maxFactorsIdx := lf-1;
end;

procedure Check(var factors:tFactors;max:Uint32);
var
i: Uint32;
begin
GetFactorList(factors,max);
write(max:10,': ');
if maxFactorsIdx>15 then
CheckDigits(factors)
else
writeln(ConvertToUint64(factors):21,' = ',ConvertToStr(factors));
for i := 0 to saveFactorsIdx do
factors[i] := savefactors[i];
end;

var
max: Uint32;
BEGIN
Init_Primes;

max := 2;
repeat
check(factors,max);
max *=10;
until max > MAX_LIMIT;

writeln;
For max := 10 to 20 do // < MAX_UINT64
check(factors,max);
{\$IFDEF WINDOWS}
{\$ENDIF}
END.
```
Output:
```TIO.RUN Real time: 1.161 s User time: 1.106 s Sys. time: 0.049 s CPU share: 99.49 %
2:                     2 = 2
20:             232792560 = 16*9*5*7*11*13*17*19
200:  has         46 factors and         90 digits
3372935888329262646..8060677390066992000
2000:  has        303 factors and        867 digits
1511177948774443153..3786415805463680000
20000:  has       2262 factors and       8676 digits
4879325627288270518..7411295098112000000
200000:  has      17984 factors and      86871 digits
3942319728529926377..9513860925440000000
2000000:  has     148933 factors and     868639 digits
8467191629995920178..6480233472000000000
{ at home
20000000:  has    1270607 factors and    8686151 digits
1681437413936981958..6706037760000000000
200000000:  has   11078937 factors and   86857606 digits
2000000000:  has   98222287 factors and  868583388 digits
}
10:                  2520 = 8*9*5*7
11:                 27720 = 8*9*5*7*11
12:                 27720 = 8*9*5*7*11
13:                360360 = 8*9*5*7*11*13
14:                360360 = 8*9*5*7*11*13
15:                360360 = 8*9*5*7*11*13
16:                720720 = 16*9*5*7*11*13
17:              12252240 = 16*9*5*7*11*13*17
18:              12252240 = 16*9*5*7*11*13*17
19:             232792560 = 16*9*5*7*11*13*17*19
20:             232792560 = 16*9*5*7*11*13*17*19
```

Perl

```#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/Smallest_multiple#Raku
use warnings;
use ntheory qw( lcm );

print "for \$_, it's @{[ lcm(1 .. \$_) ]}\n" for 10, 20;
```
Output:
```for 10, it's 2520
for 20, it's 232792560
```

Phix

Using the builtin, limited to 253 aka N=36 on 32-bit, 264 aka N=46 on 64-bit.

```with javascript_semantics
?lcm(tagset(20))
```
Output:
```232792560
```

Using gmp

Translation of: Wren
```with javascript_semantics
include mpfr.e
procedure plcmz(integer n)
sequence primes = get_primes_le(n)
mpz res = mpz_init(1)
for i=1 to length(primes) do
integer p = primes[i], f = p
while f*p <= n do f *= p end while
mpz_mul_si(res,res,f)
end for
printf(1,"%,5d: %s\n", {n, shorten(mpz_get_str(res,10,true))})
end procedure

printf(1,"The LCMs of the numbers 1 to N inclusive is:\n")
papply({10,20,200,2000},plcmz)
```
Output:
```The LCMs of the numbers 1 to N inclusive is:
10: 2,520
20: 232,792,560
200: 337,293,588,832,926,...,677,390,066,992,000 (90 digits)
2,000: 151,117,794,877,444,...,415,805,463,680,000 (867 digits)
```

Picat

lcm/2

`lcm/2` is defined as:

`lcm(X,Y) = X*Y//gcd(X,Y).`

Iteration

```smallest_multiple_range1(N) = A =>
A = 1,
foreach(E in 2..N)
A := lcm(A,E)
end.```

fold/3

`smallest_multiple_range2(N) = fold(lcm, 1, 2..N).`

reduce/2

`smallest_multiple_range3(N) = reduce(lcm, 2..N).`

Testing

Of the three implementations the `fold/3` approach is slightly faster than the other two.

```main =>
foreach(N in [10,20,200,2000])
println(N=smallest_multiple_range2(N))
end.```
Output:
```10 = 2520
20 = 232792560
200 = 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000 = 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000```

Python

```""" Rosetta code task: Smallest_multiple """

from math import gcd
from functools import reduce

def lcm(a, b):
""" least common multiple """
return 0 if 0 == a or 0 == b else (
abs(a * b) // gcd(a, b)
)

for i in [10, 20, 200, 2000]:
print(str(i) + ':', reduce(lcm, range(1, i + 1)))
```
Output:
```10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000```

Quackery

`lcm` is defined at Least common multiple#Quackery.

```  [ 1 swap times [ i 1+ lcm ] ] is smalmul ( n --> n )

' [ 10 20 200 2000 ] witheach [ dup echo say ": " smalmul echo cr ]```
Output:
```10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000```

Raku

Exercise with some larger values as well.

```say "\$_: ", [lcm] 2..\$_ for <10 20 200 2000>
```
Output:
```10: 2520
20: 232792560
200: 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000
2000: 151117794877444315307536308337572822173736308853579339903227904473000476322347234655122160866668946941993951014270933512030194957221371956828843521568082173786251242333157830450435623211664308500316844478617809101158220672108895053508829266120497031742749376045929890296052805527212315382805219353316270742572401962035464878235703759464796806075131056520079836955770415021318508272982103736658633390411347759000563271226062182345964184167346918225243856348794013355418404695826256911622054015423611375261945905974225257659010379414787547681984112941581325198396634685659217861208771400322507388161967513719166366839894214040787733471287845629833993885413462225294548785581641804620417256563685280586511301918399010451347815776570842790738545306707750937624267501103840324470083425714138183905657667736579430274197734179172691637931540695631396056193786415805463680000```

Ring

```see "working..." + nl
see "Smallest multiple is:" + nl
n = 0

while true
n++
flag = 0
for m = 1 to 20
if n % m = 0
flag += 1
ok
next
if flag = 20
see "" + n + nl
exit
ok
end

see "done..." + nl```
Output:
```working...
Smallest multiple is:
232792560
done...
```

Verilog

Translation of: Yabasic
```module main;
integer temp, smalmul, lim;

initial begin
temp = 2*3*5*7*11*13*17*19;
smalmul = temp;
lim = 1;

while (lim <= 20) begin
lim = lim + 1;
while (smalmul % lim != 0) begin
lim = 1;
smalmul = smalmul + temp;
end
end

\$display(smalmul);
\$finish ;
end
endmodule
```
Output:
`232792560`

Wren

Library: Wren-math
Library: Wren-big
Library: Wren-fmt

We don't really need a computer for the task as set because it's just the product of the maximum prime powers <= 20 which is : 16 x 9 x 5 x 7 x 11 x 13 x 17 x 19 = 232,792,560.

More formally and quite quick by Wren standards at 0.017 seconds:

```import "./math" for Int
import "./big" for BigInt
import "./fmt" for Fmt

var lcm = Fn.new { |n|
var primes = Int.primeSieve(n)
var lcm = BigInt.one
for (p in primes) {
var f = p
while (f * p <= n) f = f * p
lcm = lcm * f
}
return lcm
}

System.print("The LCMs of the numbers 1 to N inclusive is:")
for (i in [10, 20, 200, 2000]) Fmt.print("\$,5d: \$,i", i, lcm.call(i))
```
Output:
```The LCMs of the numbers 1 to N inclusive is:
10: 2,520
20: 232,792,560
200: 337,293,588,832,926,264,639,465,766,794,841,407,432,394,382,785,157,234,228,847,021,917,234,018,060,677,390,066,992,000
2,000: 151,117,794,877,444,315,307,536,308,337,572,822,173,736,308,853,579,339,903,227,904,473,000,476,322,347,234,655,122,160,866,668,946,941,993,951,014,270,933,512,030,194,957,221,371,956,828,843,521,568,082,173,786,251,242,333,157,830,450,435,623,211,664,308,500,316,844,478,617,809,101,158,220,672,108,895,053,508,829,266,120,497,031,742,749,376,045,929,890,296,052,805,527,212,315,382,805,219,353,316,270,742,572,401,962,035,464,878,235,703,759,464,796,806,075,131,056,520,079,836,955,770,415,021,318,508,272,982,103,736,658,633,390,411,347,759,000,563,271,226,062,182,345,964,184,167,346,918,225,243,856,348,794,013,355,418,404,695,826,256,911,622,054,015,423,611,375,261,945,905,974,225,257,659,010,379,414,787,547,681,984,112,941,581,325,198,396,634,685,659,217,861,208,771,400,322,507,388,161,967,513,719,166,366,839,894,214,040,787,733,471,287,845,629,833,993,885,413,462,225,294,548,785,581,641,804,620,417,256,563,685,280,586,511,301,918,399,010,451,347,815,776,570,842,790,738,545,306,707,750,937,624,267,501,103,840,324,470,083,425,714,138,183,905,657,667,736,579,430,274,197,734,179,172,691,637,931,540,695,631,396,056,193,786,415,805,463,680,000
```

XPL0

```int N, D;
[N:= 2*3*5*7*11*13*17*19;
D:= 1;
repeat  D:= D+1;
if rem(N/D) then
[D:= 1;  N:= N + 2*3*5*7*11*13*17*19];
until   D = 20;
IntOut(0, N);
]```
Output:
```232792560
```

Yabasic

Translation of: XPL0
```// Rosetta Code problem: http://rosettacode.org/wiki/Smallest_multiple
// by Galileo, 05/2022

M = 2*3*5*7*11*13*17*19
N = M
D = 1
repeat
D = D + 1
if mod(N, D) D = 1 : N = N + M
until D = 20
print N```
Output:
```232792560
---Program done, press RETURN---```