I want to prove a statement, let $p\_{1}$ and $p\_{2}$ be distinct odd primes such that $(p\_{1}-1)|(p\_{2}-1)$. Let $m=p\_{1}p\_{2}$.

Prove the equation $[x]^{p\_{2}}+[x]-[1]=[0]$ has a unique solution in $\mathbb{Z}\_{m}$. [x] is a congruence class.

I think I can use the Chinese Remainder Theorem, but how do I prove uniqueness?