Re: [PATCH v2] lib: gcd: prevent possible div by 0

[Nick Bowler]

On 2012-09-12 12:36 -0700, Andrew Morton wrote:
> On Wed, 12 Sep 2012 21:20:30 +0200
> Davidlohr Bueso <dave@xxxxxxx> wrote:
> > On Wed, 2012-09-12 at 12:10 -0700, Andrew Morton wrote:
> > > On Mon, 10 Sep 2012 16:35:19 +0200
> > > Davidlohr Bueso <dave@xxxxxxx> wrote:
> > > 
> > > > Account for all properties when a and/or b are 0:
> > > > gcd(0, 0) = 0
> > > > gcd(a, 0) = a
> > > > gcd(0, b) = b
[...]
> I'm scratching my head a bit at the patch though.  What does gcd(0, 13)
> mean?  That 0 can be divided by 13 zero times, which is an integer
> result?  I wonder why any non-buggy code would do that....

The number-theoretical definition of gcd(a, b) on the integers, leaving
aside the case where a and b are both 0, are defined as the greatest
integer which divides both a and b.

An integer x divides y if there exists an integer M such that x*M
equals y.

Observe that all integers divide zero (since we can set M to 0, and
x*0 = 0 for any x).  So it's easy to see that the result of gcd(x, 0)
and gcd(0, x) must be |x|.

The case of gcd(0, 0) is tricky.  Clearly, as all integers divide zero,
none of these can be the greatest one.  So this is normally treated as a
special case, defined to be 0 by convention, as this makes the use of
gcd "nicer" in other areas of mathematics.

Cheers,
-- 
Nick Bowler, Elliptic Technologies (http://www.elliptictech.com/)

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