Question

How many zeros are there at the end of 100!?

Answer 1

`24`

Explanation:

There are plenty of factors `2` in `100!`, so the question is how many factors `5` are there?

`100!` has `100/5=20` terms divisible by `5^1`, namely `5, 10, 15, 20,..., 100`

It has `100/25 = 4` terms divisible by `5^2`, namely `25, 50, 75, 100`.

So there are a total of `20+4 = 24` factors `5` in `100!`.

Hence `100!` is divisible by `10^24` and no greater power of `10`.

So `100!` ends with `24` zeros.

A computer tells me that:

`100! = 93,326,215,443,944,152,681,699,238,856,266,700,490,715,`
`968,264,381,621,468,592,963,895,217,599,993,229,915,608,941,463,`
`976,156,518,286,253,697,920,827,223,758,251,185,210,916,864,000,`
`000,000,000,000,000,000,000`