Let $\mathbb{N}=\{1,2,3,\dots\}$ and let $\mathbb{Z}$ denote the ring of integers. Let $\varphi: \mathbb{N} \to \mathbb{N}$ be Euler's totient function, defined for $n \in \mathbb{N}$ by

\begin{align*} \varphi(n)=\left|\{k \in \mathbb{N} : 1 \le k \le n,\ \gcd(k,n)=1\}\right|, \end{align*}

where $\gcd(r,s)$ denotes the greatest positive common divisor of $r,s \in \mathbb{N}$. For $r,s \in \mathbb{Z}$, write $r \mid s$ to mean that there exists $q \in \mathbb{Z}$ such that $s=rq$. Let $p \in \mathbb{N}$ be prime, meaning $p>1$ and the only positive divisors of $p$ are $1$ and $p$, and let $a \in \mathbb{N}$ with $a \ge 1$. Then

\begin{align*} \varphi(p^a)=p^a-p^{a-1}=p^a\left(1-\frac{1}{p}\right). \end{align*}