Many properties of $\mathbb{Z}/n$ can be broken down to properties of $\mathbb{Z}/p^i$ using the Chinese Remainder Theorem. Here is an example that I have no idea how to prove otherwise:
IMO Shortlist 1997 problem 15 is equivalent to proving that if some given integer is a square residue modulo $n$ and a cubic residue modulo $n$ at the same time, then it is a $6$-th power residue modulo $n$ as well. More generally, if $n$, $u$, $v$ are three positive integers, and some given $a\in\mathbb{Z}/n$ is both a $u$-th power and a $v$-th power in $\mathbb{Z}/n$, then $a$ is a $\mathrm{lcm}\left(u,v\right)$-th power in $\mathbb{Z}/n$.