Suppose that $m$ and $n$ are coprime positive integers and that the congruences $x^2 \equiv a \pmod m$, $y^2 \equiv a \pmod n$ have $s$ and $t$ solutions in $x$ and $y$ respectively. Prove that the congruence $z^2 \equiv a \pmod{mn}$ has $st$ solutions.
Not sure how to really go about proving this.