Group-theoretic Interpretation

The set of invertible elements of $\mathbb{Z}/n\mathbb{Z}$ is a group

$$(\mathbb{Z}/n\mathbb{Z})^* = \{ [a] \in \mathbb{Z}/n\mathbb{Z}: \gcd(a,n) = 1\}.$$

This group has order  $\varphi (n)$. Theorem 4.3 asserts that the order of an element of $(\mathbb{Z}/n\mathbb{Z})^\times$ divides the order $\varphi (n)$ of $(\mathbb{Z}/n\mathbb{Z})^\times$. This is a special case of the more general theorem that if $G$ is a finite group and $g\in G$, then the order of $g$ divide $\char93 G$.