Life is A Test 2: Test Harder (Quite A Bit Harder)

by Brony_of_Brody

The Answer 41

There are 24 zeros at the end of 100!.

What this question is basically asking is "How many times is 100! able to be divided by 10?". Each zero represents one time, so any number with 00 at the end can be divided by 10 two times.

So let's go through each one. 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 Are divisible by 10, so that makes a total of 11 zeroes at the end. But wait. What about the numbers that can produce numbers with zeros at the end that don't end in 0? Such as, for example, 4 x 5 = 20?

Well, there is a way to work out how many such combinations there are. Let's break down 8 x 5 as an example. This can be expressed as (2 x 2 x 2) x 5, which can be rearranged as (2 x 2) x (2 x 5) = (4 x 10).

Likewise, something like 4 x 15 can be expressed as (2 x 2) x (3 x 5) = (6 x 10). Basically, when two numbers that don't end in zero multiply to make a number that does, it must involve a 2 and a 5 in the multiplicative breakdown. So the puzzle can be rephrased as something simpler: "5 divides into 100! how many times?" We can lay out the numbers as such counting upwards:

5, 10, 15, 20, 25...90, 95, 100.

All these terms divide by 5 once, except for 25, 50, 75 and 100, which you can divide by 5 twice, for a total of twenty-four times. So that's our answer.