proof question?

Question

Prove that the linear equation cx + dy = e with c,d,e are integers and c,d aren't equal to zero has a solution in integers (x,y) iff gcd(c,d) | e.

Answer 1

if it has a solution then obvoius gcd(c,d)|e

now suppose g = gcd(c,d)|e , then c=c'g, d=d'g, e= e'g
so c'g x+ d'gy = e'g
so
c'x+d'y = e'
But gcd(c',d')=1 by a well-known result there exist x',y' such that:
c'x'+d'y' =1. Now take x=x'e and y=y'e and you have the result