elementary number theory - If $N$ is a multiple of $100$, $N!$ ends with $left(frac{N}4-1 right)$ zeroes.

Did certain questions about factorials, and one of them got a reply very interesting that someone told me that it is possible to show that

If $N$ is a multiple of $100$, $N!$ ends with $\left(\frac{N}4-1 \right)$ zeroes.

Sought such a demonstration, not found when trying to do there, I thought of induction, but did not fit, I thought of several ways, but did not get any progress, would like to know how to do, or to see a demonstration of this.

I thought about trying to calculate the following series

$$\sum_{k=0}^{\infty} \left[\frac{n}{5^k}\right]$$ only that I was in doubt because it is using the quotients $\left[\frac{n}{5^k}\right]$