Question

why does 100! have 24 zeroes

Answer 1

prime factorize : 100! = 2^m * 3^n * 5^r....

to produce a 0 at the end, you will need a 2 and a 5
since there will be plenty of 2's compared to 5's, we need to check oly how many 5's area available to multiply wid 2

Answer 2

number of 5's = 100/5 + 100/5^2 = 20 + 4 = 24

Answer 3

prime factorization will have : 5^24
and 2 power will be larger than that.

so there will be exactly 24 0's in the tail of in 100!

Answer 4

My problem is to approach it with a list view of the numbers. Like this.

The number of zeroes found in the ten's - that's 11 because there are two zeroes in 100
The number of 5's found in the 5's - that's ten
The number of 5's found in 5^2 numbers - that's 3

Which comes to the same result.

Thank you for a more rigorous approach

Answer 5

looks good :)
both methods are one and same actually !