Show that if $\gcd(a, b)\mid c$, then the equation $ax + by = c$ has infinitely many integer solutions for $x$ and $y$. Show that if $\gcd(a, b)\mid c$, then the equation $ax + by = c$ has infinitely
many integer solutions for $x$ and $y$.
I understand that if there is one, solution for $ax+by =c$, then there are infinitely many solutions, just because you can solve it in different ways. However, I am not sure how to show this in a proof format. 

Euclidean algorithm says there is at least one solution.
if $ax + by = c$ then $a(x - b/gcd(a,b)) + b(y + a/gcd(a,b)) = ax + by = c$.  Inductively there are infinitely many further solutions.
