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What is the highest power of 2 dividing 100!

This is what I have so far:

50 multiples of 2

25 multiples of 4

12 multiples of 8

6 multiples of 16

3 multiples of 32

1 multiple of 64

EDIT: giving a highest power of 2^97

am I missing anything here?

4 Answers 4

2

Your list of multiples is correct. But then the total is 97, not 2^97. But good to see some work.

  • 1
    There are 97 factors of 2 in 100, but saying that 2^97 is the largest power of 2 that divides 100! is then correct. I believe that you are taking a different meaning of the word "power" than intended, as I elaborated on in a comment on Log2's answer.2010-10-18
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You are not missing anything. Using WolframAlpha to factor 100! into primes, we get the answer: 2^97 × 3^48 × 5^24 × 7^16 × 11^9 × 13^7 × 17^5 × 19^5 × 23^4 × 29^3 × 31^3 × 37^2 × 41^2 × 43^2 × 47^2 × 53 × 59 × 61 × 67× 71 × 73 × 79 × 83 × 89 × 97 (239 factors, 25 distinct)

therefore, the highest power of 2 that divides 100! is 2^97.

100!/2^97 = 588971222367687651371627846346807888288472382883312574253249804256440585603406374176100610302040933304083276457607746124267578125

  • 0
    Good point, but it's also even worse because I think that muad was objecting to Ross's use of "power" to mean "exponent", but it might not have been clear to either one what the others' objections were.2010-10-19
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How to solve this in the easiest way possible...

keep dividing 100 by 2 till you get a value < 2

100 / 2 = 50

50 / 2 = 25

25 / 2 = 12 (forget about the remainder)

12 / 2 = 6

6 / 2 = 3

3 / 2 = 1

now just add up all the quotients 50 + 25 + 12 + 6 + 3 + 1 = 97

the same logic can be applied to find the highest power of any number x that divides n! completely or evenly.

  • 0
    To the OP: I have upvoted this answer, but you may want to add that the formula you're using is that I mention, this would remove the mystery of forgetting the remainder, apart from improving the answer substantially.2012-05-01
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@Adrián Barquero's link is dead, and I don't have the rep to comment, so here's the archive link from 2008:

http://web.archive.org/web/20080430101233/http://planetmath.org/encyclopedia/PrimePowerDividingAFactorial.html

  • 0
    Also, your link does unfortunately not render the latex.2017-02-13