Let us say that an integer $k$ where $0< k < m$ is a zero-divisor mod $m$ if $kn \equiv 0 \pmod{m}$ for some $n$ with $0 < n < m$.
Prove the following: If $m$ is prime then no integer $k$ is a zero-divisor mod $m$.
Homework question; I think I'm on my way to the solution?
So $kn/m$ needs to be an integer. Since $k < m$ and $n < m$, we know that neither $k$ nor $m$ is equal to $m$. Furthermore, $k > 0$ and $n > 0$, therefore $k n$ is not equal to zero. Thus $k n > 0$ and $k n$ is not divisible by $m$ (as $m$ is a prime) and therefore, there exists no integer $k$ that is a zero-divisor mod $m$.
Is that a valid proof for the question at hand? If not, what is?
