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

The result follows from Equal Powers of Finite Order Element.