Number Theory – Zeros in the Decimal Representation of k!

Question

I'd like a hint for the question:

For how many positive integers $k$ does the ordinary decimal representation of the integer $k!$ end in exactly $99$ zeros?

Thanks.

Answer 1

Hint. A number ends in $0$ if and only if it is a multiple of $10$. A number ends in two zeroes if and only if it is a multiple of $100=2^2\times 5^2$. A number ends in three zeroes if and only if it is a multiple of $1000 = 2^3\times 5^3$. Etc. So, e.g., a number ends in exactly two zeroes if and only if it is a multiple of $100$ but not of $1000$.