jmc

Since the ratio of $A$ to $B$ is $3:4$, there is an integer $k$ for which $A=3k$ and $B=4k$. Moveover, $k$ is the greatest common divisor of $A$ and $B$, since 3 and 4 are relatively prime. Recalling the identity $\mathrm{lcm}[A,B]\cdot \gcd (A,B)=AB$, we find that $120k=(3k)(4k),$ which implies $k=120/12=\boxed{10}$.