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/
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