Trailing Zeros and Zero Factorial

Date: 04/07/98 at 00:57:09
From: mark lopez
Subject: College math (mat142)

How many trailing zeros are there for 100! ? 

I found that 69! = 1.711224524 x 10^98. This is the highest my 
calculator will go. I then started working downward from a 100 to 70. 
From this I came up with 5.453767996 x 10^59.

Do I multiply 98 by 59 to come up with answer, or do I add the two 
numbers?  If you can help me with this, I would greatly appreciate it.

Thank you.

Date: 04/07/98 at 17:05:37
From: Doctor Rob
Subject: Re: College math (mat142)

The trick here is not to calculate 100! on your calculator (which only
gives you ten digits of accuracy), but to figure out how high a power 
of 10 goes into 100! evenly. For every trailing zero, there is a power 
of 10 that divides 100! evenly. In order to do that, since 10 = 2*5, 
we need to figure the highest powers of 2 and 5 dividing 100! and take 
the lesser of the two exponents. (Why?)

Consider what happens when we multiply together 1*2*3*4*5*6*..., 
starting with the lowest numbers first. Every fifth number, starting 
with 5, is divisible by 5. That gives you 100/5 = 20 factors of 5 in 
100!. But there are more. Every 25th number, starting with 25, has an 
extra factor of 5 beyond the ones already counted. That gives you 
100/25 = 4 more factors of 5 in 100!. To get a third factor of 5 from 
a single number, it has to be a multiple of 125, and no number <= 100 
is, so that is all. The answer is:

     [100/5] + [100/5^2] + [100/5^3] + ...
          = 20 + 4 + 0 + ...
          = 24

Here [x] means the integer part of x, or the greatest integer not 
exceeding x. (Why do we use this [x] instead of x?) All the terms from 
some point on will be zero, so this is a finite sum.

Now for 2's (or any other prime number), the same analysis holds. The
answer for the highest power of 2 dividing 100! is

     [100/2] + [100/2^2] + [100/2^3] + [100/2^4] + ...
          = 50 + 25 + 12 + 6 + 3 + 1 + 0 + 0 + ...
          = 97
 
The smaller of the two is 24, so the highest power of 10 dividing 100! 
is 10^24, so 100! ends with 24 zeroes.

The same analysis works for any factorial n! and any prime p. The 
highest power of p dividing n! is:

     [n/p] + [n/p^2] + [n/p^3] + [n/p^4] + ...

-Doctor Rob,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/