## Divisibility of factorials

### Problem 549

The smallest number m such that 10 divides m! is m=5.

The smallest number m such that 25 divides m! is m=10.

Let s(`n`) be the smallest number m such that `n` divides m!.

So s(10)=5 and s(25)=10.

Let S(`n`) be ∑s(`i`) for 2 ≤ `i` ≤ `n`.

S(100)=2012.

Find S(10^{8}).