$hm + qn = 0$ (mod $mn$) implies $h,q=0$

Question

If we know that $m$ and $n$ are coprime integers then $hm + qn = 0$ (mod $mn$) must mean that integers $h$ and $q$ must be equal to zero. However, I think I'm missing something here. How does one prove this fact?

(We know that $0 \leq h < n$ and $0 \leq q < m$). We could probably say that $hm = -qn$ (mod $mn$). But what to do next?

Answer 1

Once you know $hm+qn\equiv 0\pmod{mn}$ it is easy to find $hm\equiv 0\pmod n$. Now multiply both sides of this by $m^{-1}$.