Problem #PRU-60514

Problem

Numbers $a,b,c$ are integers with $a$ and $b$ being coprime. Let us assume that integers $x_0$ and $y_0$ are a solution for the equation $ax+by=c$.

Prove that every solution for this equation has the same form $x=x_0+kb$, $y=y_0-ka$, with $k$ being a random integer.