# Thread: [SOLVED] How many 0's at the end of 100!

1. ## [SOLVED] How many 0's at the end of 100!

How many 0's are at the end of 100! ?
Can someone check if this is correct? Thanks.

My reasoning: numbers that produce 0's at the end are 2, 5 and any other integers 2^r < 100 , 5^s < 100
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
5^1 = 5
5^2 = 25
r + s = 1 + 2 + 3 + 4 + 5 + 6 + 1 + 2 = 24
There are 24 0's at the end of 100!

2. Hello, VENI!

Amazing . . . you did it all wrong and got the right answer!

How many 0's are at the end of 100! ?
It has nothing to do with the powers-of-2.

Question: how many factors-of-5 are in 100-factorial?

Reasoning: Each factor-of-5, paired with any even number, will produce a zero at the end.

Since half the numbers are even, there are plenty of even numbers available.

So how many factors-of-5 are there?

Well, every fifth number is a multiple of 5.
. . So there are: .$\displaystyle \frac{100}{5} \,=\,20$ of them.

But some of them are multiples of $\displaystyle 5^2=25$, which provide an extra factor of 5.
. . And there are: .$\displaystyle \frac{100}{25} = 4$ of them.

Hence, there are: .$\displaystyle 20+4\:=\:24$ factors-of-5 in 100-factorial.

Therefore, it ends in 24 zeros.

3. Haha, what a coincidence.

4. Originally Posted by VENI
How many 0's are at the end of 100! ?
Some online articles explain the general method, so you can find the number of zeroes for any factorial.

Have fun!