Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order

Theorem
Let $C_n$ be the cyclic group of order $n$.

Let $C_n = \left \langle {a} \right \rangle$, that is, that $C_n$ is generated by $a$.

Then:
 * $C_n = \left \langle {a^k} \right \rangle \iff k \perp n$

That is, $C_n$ is also generated by $a^k$ iff $k$ is coprime to $n$.

Necessary Condition
Let $k \perp n$.

Then by Integer Combination of Coprime Integers:
 * $\exists u, v \in \Z: 1 = u k + v n$

So $\forall m \in \Z$, we have:

Thus $a^k$ generate $C_n$.

Sufficient Condition
Let $C_n = \left \langle {a^k} \right \rangle$.

That is, let $a^k$ generate $C_n$.