If $a,b,c\neq0$, prove that $ac\mid bc \iff a\mid b$

Question

How can I prove this question?

If $a,b,c\neq0$, prove that $ac\mid bc \iff a\mid b$

Please help me

Answer 1

$bc=(ac)k\iff b=ak$. Of course you can only cancel $c$ if $c$ is different to zero.${}{}{}{}{}{}{}{}{}{}{}$

Answer 2

The expression $ac \mid bc$ means that there exists an integer $n \in \mathbb Z$ such that $$bc = n(ac).$$

Since $c \neq 0$ $$b = na \iff a \mid b.$$