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### The Number of Zeros in a Factorial

```
Date: 10/01/98 at 15:29:31
From: jeff
Subject: Large factorial problem

The problem is: How many zeros come after the last non-zero digit of
20,000,000! ?

We have considered all known possibilities of numbers ending in zero,
including products of '5' and '2', and came up with: 2.9 with .9
repeating times ten to the sixth power zeros to compensate for any
overlapping numbers in our computations.

I am stumped. Can you help me?
```

```
Date: 10/01/98 at 17:31:17
From: Doctor Nick
Subject: Re: Large factorial problem

Hi Jeff -

The number of zeros in n! is determined by summing up [n/(5^i)], where
[x] = the largest integer less than or equal to x, and the sum is for
i = 1,2,3,4,.... . Eventually, n/(5^i) will be less than 1, so that
[n/(5^i)] = 0, and you can stop adding.

For example:

40! = 815915283247897734345611269596115894272000000000

and

[40/5] = 8
[40/(5^2)] = 1
[40/(5^3)] = 0

and these sum to 9 - there are 9 zeros.

The idea is that each multiple of 5 between 1 and 40 contributes a
zero to 40!. Multiples of 5^2 contribute another 1, multiples of 5^3
contribute yet another one, and so on. There are lots of multiples of
2 around, so it's the 5's that determine how many multiples of 10
(i.e. how many zeros) we get.

Now, how many multiples of 5 are there between 1 and n? Well, if n is
a multiple of 5, then there are n/5. If not, let n' be the largest
multiple of 5 less than n. Then there are n'/5 multiples of 5 between
1 and n. Since n' is one of n-1, n-2, n-3, or n-4, it follows that
n'/5 = [n/5]. We can show by the same sort of argument that the number
of multiple of 5^2 less and n is [n/(5^2)], and so on.

In your case, with n=20,000,000, we have:

i    [n/(5^i)]
1    4000000
2    800000
3    160000
4    32000
5    6400
6    1280
7    256
8    51
9    10
10    2
11    0

and these sum to 4,999,999. Thus, there are precisely 4,999,999 zeros
after the last non-zero digit of 20,000,000!.

Enjoy,

- Doctor Nick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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