Given two integers a and b, we say that a divides b (and write a|b) if there is an integer c such that ac=b

Given integers a,b, and c, we say that c is the greatest common divisor of a and b (and write c=GCD(a,b)) if:
i)c|a and c|b.
ii)For any integer d satisfying both d|a and d|b, we have c≥d.

You may assume that any two integers have a unique greatest common divisor. (A proof is here: .ps, .pdf)

So here's the problem: Given integers x,y, and a, prove that GCD(x,y)=GCD(x,ax+y).

This is a fairly straightforward exercise in verifying the definitions; we'll have a lot of proofs that look sort of like it over the course of the semester.

If you want criticism from me, you can write up your proof in the space below and submit it; I'll return it with comments at the next section meeting. (You get absolutely no credit for doing this, but the feedback might prove beneficial.)