Given that $gcd (a, b) = 1$ and integers $x_0$ and $y_0$ are solutions to $ax + by= 1$, determine an integer solution to $ax + by= c$ where $c$ is an integer.
We're given
$$ a(x_0) + b(y_0) = 1 $$
Multiplying though by $c$ we have
$$ a(cx_0) + b(cy_0) = c $$
Here $cx_0$ and $cy_0$ are integers, because $c, x_0, y_0$ are integers.
So $x=cx_0, y=cy_0$ is an integer solution of $ax + by= c$.
Note: We don't make use of $\gcd(a,b)=1$.