Let $(G, \cdot)$ be a finite [cyclic group](/page/Cyclic%20Group) with identity element $e$, let $g \in G$, and let $n \in \mathbb{N}$ satisfy $G=\langle g \rangle$ and $|G|=n$. For every $k \in \mathbb{Z}$, with $\gcd(n,k)$ denoting the positive greatest common divisor of $n$ and $k$, the order of the element $g^k \in G$ is