Order of Power of Group Element

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $g \in G$ be an element of $G$ such that:
 * $\left|{g}\right| = n$

where $\left|{g}\right|$ denotes the order of $g$.

Then:
 * $\forall m \in \Z: \left|{g^m}\right| = \dfrac n {\gcd \left\{{m, n}\right\}}$

where $\gcd \left\{{m, n}\right\}$ denotes the greatest common divisor of $m$ and $n$.