Chris Wuthrich has mentioned that the structure of the group $({\bf Z}/p^n{\bf Z})^\times$ can be easily determined by $p$-adic methods. These local methods are really indispensable for determining the structure of the group $({\mathfrak o}/{\mathfrak p}^n)^\times$ in general, where ${\mathfrak o}$ is the ring of integers in a number field and ${\mathfrak p}\subset{\mathfrak o}$ is a prime ideal. See Chapter 15 of Hasse's Number Theory.
As a related application, consider Wilson's theorem ($(p-1)!\equiv-1\pmod p$ for a prime number $p$). More generally, Gauss (Disquisitiones, $\S$78) determined the product of all elements of $({\bf Z}/a{\bf Z})^\times$ for every $a>0$. There are now two possibilities, $+1$ and $-1$, and Gauss proves that the product is $-1$ precisely when $a$ is $4$, or $p^m$, or $2p^m$ for some odd prime $p$ and integer $m>0$.
For an ideal ${\mathfrak a}\subset{\mathfrak o}$ in the ring of integers of a number field, what is the product of all elements in $({\mathfrak o}/{\mathfrak a})^\times$ ? There are now four possibilities, and one can say which one occurs when. This turns out to be a result of Laššák Miroslav (2000) but I had a lot of fun a few years ago giving a simple $p$-adic proof in JTNB (or arXiv).
Addendum. Laššák's paper is available online now-a-days.