jmc

Since the ratio of $A$ to $B$ is $3:7$, there is an integer $k$ for which $A=3k$ and $B=7k$. Moveover, $k$ is the greatest common divisor of $A$ and $B$, since 3 and 7 are relatively prime. Recalling the identity $\mathrm{lcm}[A,B]\cdot \gcd (A,B)=AB$, we find that $1575k=(3k)(7k),$ which implies $k=1575/21=\boxed{75}$.