Earlier today I posted the following video, in which I asked Google Assistant to calculate the factorial of 100.

The factorial of 100 is the multiplication 100 x 99 x 98 x … x 3 x 2 x 1 in which 100 is multiplied by each integer below it.

The answer has 158 digits. However, Google Assistant’s valiant efforts don’t get all the numbers right.

Today’s riddle was:

*How many zeros is the factorial of 100 really have at the end?*

**Solution: **

[I will use the mathematical symbol ‘!’ for factorial below. Thus the factorial of 100 is also written 100!.]

I mentioned in the original post that if a number has a trailing zero it is divisible by 10. What we need to do here is figure out how many times 10 divides into the number 100 x 99 x 98 x … x 3 x 2 x 1.

Let’s start: 10 divides once each into 10, 20, 30, 40, 50, 60, 70, 80, 90 and twice into 100, which means there must be at least 11 zeros at the end of 100 !.

However, it is possible to multiply two numbers that do not end in 0 to create one that does. For example, 8 x 5 = 40. How do you count all the times that count in the distribution of 100! multiply together to get a number divisible by 10?

The clue is to realize that 10 = 2 x 5. And that every time two numbers multiply to create a number divisible by 10, there must be a 2 and a 5 involved.

For example 8 x 5 = 2 x 2 x 2 x 5 = (2 x 2) x (2 x 5) = 4 x 10 = 40.

Thus, we can reformulate our task by finding all instances of (2 x 5) in 100!. In other words, we need to break down each number from 1 to 100 into its factors and see how many times 5 and 2 appear.

How many times does 5 appear as a factor in numbers from 1 to 100? Well, counting upwards by 5, we get 5, 10, 15, 20… 90, 95, 100. These 20 numbers have five as a factor. In fact 25, 50, 75 and 100 have 5 as a factor twice. So the number of times 5 appears as a factor is 24.

We can quickly see that 2 appears as a factor at least 24 times (just count the even numbers), so the total number of times (2 x 5) appears in 100! must be 24. The number of zeros at the end of 100! is 24 years old.

For those interested, 100! in all its glory is:

9332621544394415268169923885626670049071596826438162146859296389521759999322991756089414639761223758251185210916864000000000000000

And There you go!

*I install a puzzle here every two weeks on a Monday. I’m always on the lookout for great puzzles. If you want to suggest one, write to me.*

*I’ve authored several puzzle books, including the most recent Language Lover’s Puzzle Book. I also give school lectures on math and puzzles (online and in person). If your school is interested, please contact us.*