Proofing the existence of a non-zero congruence class

Question

Let $m\ge2$ be an integer. Show that if there is an integer $a$ such that $\gcd(a,m)=d\not=1$, then there exists a non-zero congruence class $[x]$ in $\mathbb{Z}_m$, such that $[a]\cdot[x]=[0]$.

I really don't know how to approach this. Any help to get started would be really appreciated..

Answer 1

Hint: If $m = dk$ for some $k \in \Bbb{Z}$, try $[a] \cdot [k]$.