Suppose that for some $m\ge1$ and $a$ and $b$ with $\gcd(a,m)=\gcd(b,m)=1$ we have $\operatorname{ord}_ma=k$ and $\operatorname{ord}_mb=l$ where $\gcd(k,l)=1$. Prove that $\operatorname{ord}_m(ab)=kl$.
What I have to go on:
If $(ab)^s \equiv 1 \pmod m$ for some $s \ge 1$, raise both sides of this congruence to the power $k$ and see what this tells you about $s$.