If the least common multiple of $A$ and $B$ is $120$, and the ratio of $A$ to $B$ is $3:4$, then what is their greatest common divisor?
To find the greatest common divisor of $A$ and $B$, I need to factorize them into their prime factors and look for the common ones. Since $A$ and $B$ have a ratio of $3:4$, I can write them as $3k$ and $4k$ for some positive integer $k$. Then, their least common multiple is $12k$, and I know this is equal to $120$. Therefore, $k=10$, and $A=30$ and $B=40$. Now, I can factorize $30$ as $2 \times 3 \times 5$ and $40$ as $2^3 \times 5$. The common factors are $2$ and $5$, so the greatest common divisor of $A$ and $B$ is $2 \times 5 = 10$. 