What is the least positive integer $n$ such that 1560 divides $n!$?
To find the least positive integer $n$ such that 1560 divides $n!$, I need to factorize both 1560 and $n!$. 1560 can be written as $2^3 \cdot 3^2 \cdot 5 \cdot 13$. This means that $n!$ must have at least three factors of 2, two factors of 3, one factor of 5, and one factor of 13. The smallest $n$ that satisfies this condition is 13, since 13! has exactly one factor of 13, and more than enough factors of 2, 3, and 5. If I choose a smaller $n$, then $n!$ would not have a factor of 13, and 1560 would not divide it. Therefore, the least positive integer $n$ such that 1560 divides $n!$ is 13.

