Hello , i got a pretty difficult task given by my Proffesor. I need to find how many 0's (in the row) at the end of number 2014! are there. If i wasn't clear enough , there's an example: number 10! = 3628800 - 2 zeros at the end in the row , 100! got 24 zeros. I doubt there is good "mathelhead" out here but its worth a shot. Have a nice day.

Guest Oct 2, 2014

#1**+5 **

Here's the proceedure for finding the answer

- Take the number that you've been given the factorial of.
- Divide by 5; if you get a decimal, truncate to a whole number.
- Divide by 5
^{2}= 25; if you get a decimal, truncate to a whole number. - Divide by 5
^{3}= 125; if you get a decimal, truncate to a whole number. - Continue with ever-higher powers of 5, until your division results in a number less than 1. Once the division is less than 1, stop.
- Sum all the whole numbers you got in your divisions. This is the number of trailing zeroes.

So we have

2014 / 5 = 402.8 = 402

2014 / 5^{2} = 80.56 = 80

2014 / 5^{3} = 16.112 = 16

2014 / 5^{4} = 3.2224 = 3

2014 / 5^{5} = .644 and we can stop

So...summing 402 + 80 + 16 + 3 we have 501 trailing zeroes

You can read about why this proceedure works here : http://www.purplemath.com/modules/factzero.htm

CPhill
Oct 2, 2014

#2**+5 **

My way of looking at this is:

every number that ends in a 2 multiplied by a number ending in a 5 adds one more zero to the number of zeroes;

also multiplying by a number that ends in a zero adds another zero.

This means that 10! has two zeroes because it has 2 x 5 x 10; one zero for the 2 x 5 and another zero for the 10.

But this means: 1 through 10 adds 2 zeroes; so does 11 through 20; 21 through 30; up through 91 through 100

for a total of 20 zeros

But I forgot about 25, since it contains 2 5's as factors, it will add 2 zeros not just one (there are a lot of extra 2's in the other numbers); so here is an extra zero.

Then, there is also, 50, 75, and 100; here are three more zeros.

OK, that's how you can get 24 zeros in 100!.

Can you use this to finish the analysis for 2014! ?

geno3141
Oct 2, 2014