Show that if $gcd(a,b)|c$ then $ax+by=c$ has infinitely many solutions.

Question

Show that if $gcd(a,b)|c$ then $ax+by=c$ has infinitely many solutions.

I know that in order to find the solutions for $x$ and $y$, you can use the equations $x= x_o + (b/d)t$ and $y=y_0 - (a/d)t$ where $d=gcd(a,b)$ and $ t \in Z$. What I don't know is how to show this using some sort of proof.

Answer 1

Using the Extended Euclidean Algorithm you can construct one solution to this equation. Call it $(x_0,y_0)$.

Then clearly, $(x_0+n \cdot b,y_0-n \cdot a)$ is a solution for any integer $n$. This gives you an infinite number of solutions.