For how many positive integers n does n! end in exactly 5 0s?

Answer 1

we get a zero when we multiply be a number ending in either 2 and 5 or a 10
at the end of 5! --- one zero
at the end of 10! -- two zeros
at the end of 15! -- 3 zeros
at the end of 20! -- 4 zeros
at the end of 25! -- 5 zeros

e.g.
10! = 10x9x8x7x6x5x5x3x2x1
= 2x5x3x3x2x2x2x7x3x2x5x2x2x3x2
= (2^8 x 5^2) x 3^4 x 7
= 6400 x 567 --- ending in 2 zeros