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# Number 2014! , How many 0's it got in the row at the end of that number ?

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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

#3
+17711
+5

CPill's answer is far more complete than mine, with exactly the correct procedure. But, in the problem, he transposed 2014 into 2104, so check the last two numbers.

geno3141  Oct 2, 2014
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#1
+84382
+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 52 = 25; if you get a decimal, truncate to a whole number.
• Divide by 53 = 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 / 52 = 80.56 = 80

2014 / 53 =  16.112 = 16

2014 / 54 = 3.2224 = 3

2014 / 55 =  .644   and we can stop

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

CPhill  Oct 2, 2014
#2
+17711
+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
#3
+17711
+5

CPill's answer is far more complete than mine, with exactly the correct procedure. But, in the problem, he transposed 2014 into 2104, so check the last two numbers.

geno3141  Oct 2, 2014
#4
+84382
0

Thanks geno......I'll fix that !!!   (It was just a mis-type .....)  ...the answers are correct.....

CPhill  Oct 2, 2014

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