Solution

Prime factorization

5! = 120

That's because 5 x 2 = 10, so there's a hidden 10 in 5!. In other words, the prime factorization of 10 is 5 x 2, so if you can determine the number of 5 x 2 combinations in 100!. If you think about it, there are many more factors of 2 than there are factors of 5 in 100!, since every other number contributes at least 1 factor of 2.

So let's focus on how many factors of 5 there are in 100! :

• 5, 10, 15, 20, ... 90, 95, 100 -- that makes 20 numbers between 1 and 100.

So the answer is 20 trailing zeros, right? Well not quite, but close. There are a few more...

Think of it this way, going back to prime factorization:

• 10 = 5 x 2
• 15 = 5 x 3
• 20 = 5 x 2 x 2
• 25 = 5 x 5

Some numbers contribute more than one factor of 5. How many numbers contribute more than one factor of 5? Which really means how many numbers include 5 x 5?

• 25, 50, 75, and 100

That gives us 24 trailing zeros.

Are there any more? Are there any numbers that contribute more than two factors of 5? Since 5 x 5 x 5 = 125, the answer is no, so 24 is the correct answer.