Corollary 1

Let $G$ be a cyclic group of order $n$ and let $d$ be a divisor of $n$. Then the number of elements of order $d$ in $G$ is equal to $\varphi(d)$. In particular, $\varphi(n)$ is the number of generators of $G$.

Since $G$ is isomorphic to $\mathbf{Z}/n\mathbf{Z}$, the number of generators of $G$ is equal to $\varphi(n)$ by Prop. 3. Let $g$ be a generator of $G$; then the elements $h$ of $G$ such that $h^d = 1$ constitute the subgroup $H$ of $G$ generated by $g^{n/d}$; this group is cyclic of order $d$ and the elements of order $d$ of $G$ are the generators of $H$, hence their number is $\varphi(d)$.