If an integer $a$ is not divisible by an integer $b$, could this imply that $b = 0$?

Question

I am working on a proof in advanced mathematics, and I believe my professor may have overlooked something. The proof is that if $d|a$, $d|b$, and $d\not|c$, then $ax+by=c$ has no integer solutions for $x$ and $y$. However, if $x$, $y$, and $c$ are $0$, that would be an integer solution.

Answer 1

$d|0$. Everything divides $0$.