Element to Power of Remainder

Theorem
Let $G$ be a group whose identity is $e$.

Let $a \in G$ have finite order such that $\left|{a}\right| = k$.

Then:
 * $\forall n \in \Z: n = q k + r: 0 \le r < k \iff a^n = a^r$

Proof
Let $n \in \Z$.

We have:
 * $n = q k + r \iff n - r = q k \iff k \mathrel \backslash \left({n - r}\right)$

The result follows from Equal Powers of Finite Order Element.