Question regarding nilpotent ideals of a ring.

Question

I am working on the following:

An ideal $N$ is called nilpotent if $N^n$ is the zero ideal for some $n\geq1$. Prove that the ideal $p\mathbb{Z}/p^m\mathbb{Z}$ is a nilpotent ideal in the ring $\mathbb{Z}/p^m\mathbb{Z}$.

I think I've constructed a valid proof but I want to verify; is $N^n$ the zero ideal iff $m | n$?

Answer 1

$N^n$ is the zero ideal if and only if $n \geq m$.