Is the gcd function symmetric?

[iamsmooth]

Just a quick theory question. I'd assume it is, but usually the bigger number goes first.

e.g. gcd(10, 5) = 2

but does gcd (5, 10) = 2?

My guess is yes.

Thanks for the help.

 

[Gigasoft]

Yes. This is apparent from the definition of the words "greatest" and "common".

 

[Legendre]

My input is probably trivial but I thought it would be nice to prove it formally.

We need : gcd(x,y) = gcd(y,x)

Suppose gcd(x,y) = d. Then x = nd and y = md for some integers n,m where gcd(n,m) = 1.

So gcd(y,x) = gcd (md,nd) = d because gcd(m,n) =1.

We have shown for all x,y that gcd(x,y) = gcd(y,x).

P.S.
Hopefully no silly errors. -_-'''