prove that every homomorphism $f: \mathbb{Q} \to \mathbb{Z}/m\mathbb{Z}$ is trivial

Question

prove that every homomorphism $f: \mathbb{Q} \to \mathbb{Z}/m\mathbb{Z}$ is trivial.

I am stuck with this problem for plenty of time but I still have no idea how I should approach it. I am thinking about prove by contradiction, i.e, suppose that there exist a non-trivial homomorphism from Q to Z/mZ then derive a contradiction. But I can't even see where the contradiction could be. Thank you.

Answer 1

$$f(x)=mf(\frac{x}{m})=0$$ since in $\mathbb{Z}_m$ the $m$th multiple of any element is zero.