I have this math problem, that I'm kind of confused on.
Consider the equation $a x + b y = c$, for some non-zero integers $a, b$ and $c$. Suppose that $x = x_1, y = y_1$ is an integer solution to the equation $ax + by = c$.
i) Show that for any $m \in \mathbb{Z}$, $$ x = x_1+ \frac{b}{\gcd(a,b)}m, y=y_1 - \frac{a}{\gcd(a,b)}m $$ will also satisfy the equation $ax + by = c$ and hence is also an integer solution.
ii) It is know that if $x, y$ is an integer solution to the homogeneous equation $a x + by = 0$, then $$ x = \frac{b}{\gcd(a,b)} k, y= - \frac{a}{\gcd(a,b)} k, \text{ for some integer } k. $$ Use this fact to show that if $x=x_2, y=y_2$ is another solution to the equation $a x + b y = c$, then $$ x_2 = x_1+ \frac{b}{\gcd(a,b)}k, y_2=y_1 - \frac{a}{\gcd(a,b)}k, \text{ for some integer } k. $$
I'm not 100% sure on how to start i). I know that $\gcd(a, b)$ can be represented as $ar+bs=\gcd(a,b)$. But I'm not sure how to go about answering this question. Thanks.