Proof: $\mathbb{Z}_n$ has zero divisors if and only if $n$ is not prime.

Question

I know this has to be proven both ways since it is an if and only if question, but I do not know how to go about this or even where to start either side of the proof. I know $[a][b]=[0]$ for there to be a zero divisor. Please help!

Answer 1

$n$ is not prime $\iff n=ab, 1\lt a,b\lt n\iff ab\cong0\pmod n\iff a$ and $b$ are zero divisors.