# Talk:Greatest common divisor

## errors in programs

A few programs would attempt to divide by zero if the 2nd argument is 0 (zero).
In that special case, the absolute value of the first argument should be returned. -- Gerard Schildberger 15:39, 17 August 2012 (UTC)

unless it is 0 --Walterpachl 12:40, 17 August 2012 (UTC)
This gets into the definition of GCD. What you are proposing here -- that 0 GCD 0 be an exceptional case -- would mean that GCD is not associative. Meanwhile, Boolean algebra/rings uses an associative definition for GCD. So I guess I do not feel motivated to adopt your definition. --Rdm 17:29, 17 August 2012 (UTC)
I hope I made it clear that it wasn't my definition, but a convention that some people use. I chose the definition [gcd(0,0=0] that didn't cause a SYNTAX condition to be raised (in REXX) and cause the program to raise an error condition, or cause it to go into error recovery. The GCD function is normally only defined for non-zero integers (some define it for only positive integers), it's the case(s) where there're arguments which are zero that are contested. -- Gerard Schildberger 21:25, 17 August 2012 (UTC)
It hurts to read that a divisor (how great or common it may be) should be zero when dividing by zero is among the worst things you can do. As for me, I rather go for undefined! --Walterpachl 20:47, 17 August 2012 (UTC)
And, yet, this definition -- where 0 == gcd(0,0) -- is the basis of the original formulation of boolean algebra. It hurts that this bit of history, of math, of logic, of the terminology we use -- is actively being lost to redefinition (and perhaps an unwillingness to think things through).
The problem with dividing a number which is not zero, by zero is that you get a zero result when you multiply zero by any number. So, obviously, the issue of dividing zero by zero is different -- here, the problem is not that there's no valid answer. When you divide zero by zero, the problem is that all answers are valid. So, 0 as the answer for 0 == GCD(0,0) has to be valid. It's not the only valid answer, but it's algorithmically simple (we do not need any exceptions in the euclidean algorithm to deal with this case) and it's syntactically simple (it lets GCD be associative).
And, this definition corresponds to the definition that Claude Shannon used for for his subset of George Boole's algebra. There are other ways of using algebraic notation with logical values, but I am reminded of this approach every time I use a "boolean value". --Rdm 13:12, 18 August 2012 (UTC)

The special case of   gcd(0,0)   is usually defined to be   0,   but some authors consider it to be undefined.   When implementing the REXX version 1 example,   the first definition (equal to zero) was chosen.   So, for that case,   │0│ = 0.

It is useful to define   gcd(0,0) = 0   and   lcm(0,0) = 0   because then the natural numbers become a complete distributive lattice with   gcd   as the   meet   operation,   and   lcm   as the   join   operation.

-- Gerard Schildberger 16:10, 17 August 2012 (UTC)

A number of examples don't show the results if either argument is negative. -- Gerard Schildberger 15:39, 17 August 2012 (UTC)

## REXX Version 1

These results seem to be wrong:

the GCD of 14 and 0 and 7 is 14 should be 7
the GCD of 0 and 7 is 0 should be 7
the GCD of 0 and 0 is 0 should be ???

correct:

the GCD of 7 and 0 is 7

--Walterpachl 12:06, 17 August 2012 (UTC)

Cases of zero have been corrected (the GCD subroutine was exiting instead of processing more arguments). The special case of gcd(0,0) can be defined to be 0, or undefined. Zero was choosen. -- Gerard Schildberger 17:08, 17 August 2012 (UTC)

And, yes, Wikipedia claims that gcd is associative. --Rdm 17:31, 17 August 2012 (UTC)

I don't understand why this is relevant! It's allso communicative.
I'll surrender and change indef(inite) to 0 as much as I dislike to read DIVISOR=0 --Walterpachl 15:17, 18 August 2012 (UTC)

## PL/I

should return a positive integer --Walterpachl 12:39, 17 August 2012 (UTC)