Prove: For a,b,c positive integers, ac divides bc if and only if a divides b

Question

Prove the biconditional statement. For $a,b,c$ positive integers, $ac$ divides $bc$ if and only if a divides $b$.

Answer 1

The two implications follow directly from definition of division, and the fact that $c$ is positive, so you can divide by it in the two sides of an equallity.

Answer 2

If $a$ divides $b$, then $b = a \cdot k$ for some integer $k$. Thus if you multiply both sides by $c$, we get the desired $bc = akc$.

On the other hand, if $ac$ divides $bc$, then $bc = akc$ for some integer $k$. Dividing both sides by $c$ yields the desired result.