When is $r = r^{2}$ in $\mathbb{Z}/p^{l}\mathbb{Z}$?

Question

When is $r = r^{2}$ in $\mathbb{Z}/p^{l}\mathbb{Z}$, where p is a prime number and l is a natural number? It obviously is the case for [0] and [1], but I am having difficulties proving that it's not possible for other r.

Answer 1

In the cyclic group $\mathbb{Z} \pmod {p^\ell}$, which elements $x$ satisfy the condition $x=x^2$. We want to solve $x \equiv x^2$, or $x(x-1) \equiv 0, \pmod {p^\ell}$. Thus $x(x-1)$ must be a multiple of $p^\ell$. But $p$ is a prime, and it is not possible for both $x$ and $x-1$ to be multiples of $p$ unless one of them is 0.