Multiplicative Inverse in Ring of Integers Modulo m

Theorem
Let $\left({\Z_m, +_m, \times_m}\right)$ be the ring of integers modulo $m$.

Then $\left[\!\left[{k}\right]\!\right]_m \in \Z_m$ has an inverse in $\left({\Z_m, \times_m}\right)$ iff $k \perp m$.