Use the Binomial Theorem to compute the order of an element in the integers mod a prime power

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.21
Let $p$ be an odd prime and $n$ a positive integer. Use the Binomial Theorem to show that $(1+p)^{p^{n-1}} = 1\ \mathsf{mod}\ p^n$ but that $(1+p)^{p^{n-2}} \neq 1\ \mathsf{mod}\ p^n$. Deduce that $1+p$ is an element of order $p^{n-1}$ in $(\mathbb{Z}/(p^n))^\times$.

Solution: TBA.