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Math Help - Number of factors

  1. #1
    Forum Admin topsquark's Avatar
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    Number of factors

    The number 12 is the smallest number with 6 distinct factors: 1, 2, 3, 4, 6, and 12. Find the smallest number that has 24 distinct factors.

    Now, I can do this given some time and trial and error. What I am interested in knowing is if there is a more "constructive" way to do this? I can't seem to think of one.

    Thanks!

    -Dan
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by topsquark View Post
    The number 12 is the smallest number with 6 distinct factors: 1, 2, 3, 4, 6, and 12. Find the smallest number that has 24 distinct factors.

    Now, I can do this given some time and trial and error. What I am interested in knowing is if there is a more "constructive" way to do this? I can't seem to think of one.

    Thanks!

    -Dan
    The numbers you mention are called highly composite numbers.

    This link might be of interest: Highly composite number - Wikipedia, the free encyclopedia

    And this link has an algorithm: http://wwwhomes.uni-bielefeld.de/ach...anuscript3.pdf
    Last edited by mr fantastic; May 6th 2008 at 05:01 AM.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by topsquark View Post
    The number 12 is the smallest number with 6 distinct factors: 1, 2, 3, 4, 6, and 12. Find the smallest number that has 24 distinct factors.

    Now, I can do this given some time and trial and error. What I am interested in knowing is if there is a more "constructive" way to do this? I can't seem to think of one.

    Thanks!

    -Dan
    Consider the prime decomposition of a number n:

     <br />
n=p_1^{k_1}...p_m^{k_m}<br />
    Then the number of factors of n is \prod_{i=1}^m(k_i+1)

    From here it is easy.

    RonL
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