State/Computing the totient of a prime power

$$\Phi(p^n) \subset R(p^n) = \{ \bar 0, \bar 1, \ldots, \overline{p^n - 1} \}.$$ We claim that $\bar x \in R(p^n) - \Phi(p^n)$ if and only if $x$ is a multiple of $p$:

By counting the multiples of $p$ between $0$ and $p^n - 1$, we find that there are $\lfloor p^n / p \rfloor = p^{n-1}$ multiples of $p$ among the residues in $R(p^n)$. Since $R(p^n)$ has $p^n$ elements, it follows that $$ \vert \Phi(p^n) \vert = \vert R(p^n) \vert - p^{n-1} = p^n - p^{n-1}.$$