I have the following problem that I'm stuck on.
Let $n>1$ be an integer and assume that the prime factorization of $n$ in the ring $\mathbb{Z}$ is $n=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{t}^{\alpha_{t}}$, where each $p_{i}^{\alpha_{i}}$ is a distinct prime in $\mathbb{Z}$. Show that an element $\overline{x}=x+n\mathbb{Z}$ in the ring $\mathbb{Z}_{n}$ is nilpotent if and only if $x$ is divisible by $p_{1}p_{2}\cdots p_{t}$.
I know the definition of nilpotent, but that's about it. I'm really confused otherwise. Thanks in advance for any help!
The element $\overline x=x+n\Bbb Z$ is nilpotent iff there's $k\in\Bbb N$ with $\overline x^k=0$. That means that $x^k+n\Bbb Z=n\Bbb Z$, that is $x^k\in n\Bbb Z$, that is $n$ divides $x^k$, that is $p_1^{a_1}\cdots p_t^{a_t}$ divides $x^k$ for some $k$.
Can you see why this is the same as saying that $p_1\cdots p_t\mid x$?