I need to prove the equivalency/inequivalency of two statements:
(1) There exists a $z \in \mathbb{Z}/n\mathbb{Z}$, $z\neq{0}$, with $z^{t} = 0$ for some $t \in \mathbb{N}$
(2) There exists a prime number $p$ with $p^{2}|n$.
Sadly, I don't have too much of an idea how to go about this. I've looked into Fermat's litte theorem which seems to be the key to this question but I can't figure it out.
Any hints where to go from there?
On
Suppose $p^2\mid n$. Then let $z=\frac np+n\Bbb Z$ and $t=2$.
Suppose $n\mid z^t$ but $n\nmid z$. By the latter, there exists a prime $p$ dividing $n$ to a higher power than $z$, whereas $n\mid z^t$ implies that $p$ divides $z^t$ to at least the same power as it divides $n$. In particular, $p$ divides $z^t$ to a higher power than $z$, hence divides $z$. In follows that $p$ divides $n$ at least quadratic.
If $n$ is square-free (i.e., $n$ is not divisible by $p^2$ for any prime number $p$), then $n=p_1p_2\cdots p_k$ for some distinct prime natural numbers $p_1,p_2,\ldots,p_k$. If $z\in\mathbb{Z}$ is such that $z^t\equiv 0\pmod{n}$ for some positive integer $t$, then $p_j\mid z^t$ for every $j=1,2,\ldots,k$, whence $p_j\mid z$ for every $j$. Ergo, $n=p_1p_2\cdots p_k=\text{lcm}\left(p_1,p_2,\ldots,p_k\right)$ divides $z$, so that $z\equiv 0\pmod{n}$.
If $n$ is not square-free, then let $z:=\text{rad}(n)$, where the radical $\text{rad}(m)$ of a positive integer $m$ is given by $$\text{rad}(m):=p_1p_2\cdots p_k\,,\text{ where }m=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}\,,$$ for pairwise distinct prime natural numbers $p_1,p_2,\ldots,p_k$ and positive integers $\alpha_1,\alpha_2,\ldots,\alpha_k$. It follows that $z\not\equiv0\pmod{n}$ and $z^t\equiv 0\pmod{n}$ for every integer $t\geq l$, where $$l:=\max_{p\text{ prime}}\,v_p(n)\,.$$ Here, $v_p$ is the $p$-adic valuation for each prime natural number $p$. Note that $l$ is the largest exponent such that there exists $z\in\mathbb{Z}$ such that $z^t\not\equiv0\pmod{n}$ for every $t=0,1,2,\ldots,l-1$ and $z^l\equiv 0\pmod{n}$.