Given $ax +by = c$ and $d= \gcd(a,b)$, where $a,b,c,d,x,y \in \mathbb{Z}$, prove the following:

A. if c does not divide d, the equation has no solution.

B. if d divide c the equation has Infinity solutions.

I need to prove that there is at least one solution, each t has a solution And does not miss a solution.

Thanks for the helpers.