How many positive integers divide $20!$ $20!$ Has nos that are multiples of $2,3,4$ and so on. However, the total number of integers is large. So, please help me.

Hint:
Use Legendre's formula:
For each prime $p\le n$,  the exponent of $p$ in the prime decomposition of $n!$ is
$$v_p(n!)=\biggl\lfloor\frac{n}{p}\biggr\rfloor+\biggl\lfloor\frac{n}{p^2}\biggr\rfloor+\biggl\lfloor\frac{n}{p^3}\biggr\rfloor+\dotsm$$
The number of prime divisors of $n!$ is then
$$\prod_{\substack{ p\;\text{prime}\\p\le n}}\bigl(v_p(n!)+1\bigr).$$
