Let $p$ be prime such that $p \not \mid a $ where $a$ is a positive integer. Prove that
$$ a^{p^n - p^{n-1}} \equiv 1 \pmod {p^n} $$
Since $p$ is prime and $p \not \mid a$, we have $\gcd(p, a)=1$, and also $\gcd(p^n, a)=1$. This means we can use Euler's Theorem,
$$ a^{\phi(p^n)} \equiv 1 \pmod {p^n} $$
Now, $\phi(p^n)=p^{n-1}(p-1) = p^n-p^{n-1}$ by Proposition (5.4), and so
$$ a^{p^n - p^{n-1}} \equiv 1 \pmod {p^n} $$