AMC 12 2020a Problem 21

How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$

$\textbf{(A) } 12 \textbf{(B) } 24 \textbf{(C) } 36 \textbf{(D) } 48 \textbf{(E) } 72$