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Date: 02/07/98 at 23:02:33
From: Kathy
Subject: Factorials

How do you find the number of zeros at the end of a number factorial?  
For example, 100! or 130!

Date: 02/08/98 at 16:41:57
From: Doctor Jaffee
Subject: Re: Factorials

Hi Kathy,

That's a very interesting question you asked. I know how I wouldn't 
calculate the number of zeros! That would be by multiplying it all 
out. A calculator wouldn't be very helpful, either. They only have 8, 
or at most 12 digits.

It occurred to me that the only way you can get a product of numbers 
to end in 0 is if one of them is a multiple of a power of 10; for 
example 20, 100, 350, or some such number, or if one is a number that 
ends in 5 and is multiplied by an even number.

So, if I want to calculate the number of zeros at the end of 100! 
I count the numbers that end in zero, namely 10,20,30,...100 (that 
will put 11 zeros at the end of our answer).  Then the numbers that 
end in 5 are 5,15,25,35,...95. (There are 10 of them, but don't forget 
that 25 and 75 are multiples of 5^2, so we have to count them twice). 
That will give us 12 more zeroes at the end of our number. And 
finally, I almost forget the number 50 has to be counted twice because 
50 is 5 x 10. For example, if you multiply 50 by an even number like 
50 x 4, you get 200, which has 2 zeroes at the end.

To summarize:  There are 10 numbers that end in zero from 1 to 100, 
               but 50 and 100 are also multiples of powers of 5, so 
               that gives us 12 zeroes.

               There are 10 numbers that end in 5 from 1 to 100,
               but 25 and 75 are also multiples of powers of 5, so 
               that gives us 12 more zeroes.

Therefore 100! must end in 24 zeroes.

You should be able to determine the number of zeroes at the end of 
130! and other factors by using this method.

I hope this has helped, and thanks for giving me the opportunity to 
work on such an interesting problem.

-Doctor Jaffee,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
Middle School Factorials

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