How many zeros

Number Theory Level 2

Find the number of trailing zeros are in the number 125 ! 125! .

30 28 29 31

Bob Yang by Bob Yang , 23, USA

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2 solutions

John Lesteя Tan
Aug 14, 2015
  • GIF (125/5) + GIF (125/25) + GIF (125/125)
  • GIF = greatest integer function
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We have to look at the power of 5 in 125! There are 25 multiples of 5 starting from 1 to 125. They will contribute 25 powers of 5. But 25 being 5^2, every multiple of it will add extra 5. So for 25, 50, 75, 100 we get 4 more 5's. And 125 is 5^3. It gives 2 more 5's. So total 25+4+2 = 31

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Please mention about the ceiling function. Thank you for solution.

Swapnil Das - 5 years, 10 months ago

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Ceiling is not used here i think. Didn't get you.

Shib Shankar Sikder - 5 years, 10 months ago

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You were lucky enough that you got a number like 125 which has 5 as its sole factor, except one and powers of the former. Please give a look at the wiki of calculation of trailing zeros of a factorial for further guidance. Thanks.

Swapnil Das - 5 years, 10 months ago

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@Swapnil Das – Yaa every new power of 5 will start a new series that will contribute more 5's we used those properties in another problem, which asked smallest number whose factorial will have 2015 trailing zeros.

Shib Shankar Sikder - 5 years, 10 months ago

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