Problem: Let $R$ be a commutative ring and let $a,b, c \in R$. Suppose that $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$.
Solution: Given that $a$ is a unit, means we can find $x \in R$ such that $ax = 1 = xa$. Let $b$ divides $c$, that is, there exists $r \in R$ such that
\begin{align*}
c = br \implies c = 1\cdot br & \implies c = xa br \\
& \implies c = (ab)\cdot (xr).
\end{align*}
Thus, $ab$ divides $c$.
On the other hand, let $ab$ divides $c$, that is, there exists $q \in R$ such that
\begin{align*}
c = ab q \implies c = b (aq).
\end{align*}
Thus, $b$ divides $c$.