Prove that the Diophantine equation $ax+by=c$ has no integer solutions if $\gcd(a,b)$ does not divide $c$

Question

Prove that the Diophantine equation $ax+by=c$ has no integer solutions if $\gcd(a,b)$ does not divide $c$ there was a hint which is use use contradiction.

Answer 1

If $d=gcd(a,b)$ then $d|a$ and $d|b$, whence $d|(ax+by)$, so it must be that $d|c$. Now take the contraposition.

Answer 2

$$\gcd(a,b)\mid ax+by=c,$$ if there exists integral solution $(x,y)$.