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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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