For all $[a]$,$[b]\in{\mathbb{Z}m}$, if $[a][b]=[0]$, then $[a]=[0]$ or $[b]=[0]$. Prove that if $S$ is true, then $m$ is prime.

Question

Let $m\in{\mathbb{N}}$ such that $m>1$. Consider the implication $S$: For all $[a]$,$[b]\in{\mathbb{Z}m}$, if $[a][b]=[0]$, then $[a]=[0]$ or $[b]=[0]$. Prove that if $S$ is true, then $m$ is prime.

How should I go about solving this implication?

Anything is appreciated, thanks!