Note: $\text{ord}_ma = k$ here is the smallest $k$ such that $a^k \equiv 1 \pmod m$, not the highest power of $m$ that divides $a$.
Is it always that case that if $\text{ord}_ma=k$ and if $\text{ord}_mb=l$ then $\text{ord}_m(ab)=kl$ should be the $\text{lcm}(k,l)$ of which is true if $\gcd(k,l)=1$?
How would I prove this?
Perhaps I could consider the case where $b$ is a multiplicative inverse of $a$?
So, $ab \equiv 1\pmod{m}$