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Math Help - Smallest number with given number of factors

  1. #1
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    Find Smallest Number For Number of Factors

    I would like to know how you would fine the smallest ordinary number with exactly a certain number of factors given, for example:

    The number of factors is 6
    The smallest number with 6 factors is 12
    12 =
    • 1 * 12
    • 2 * 6
    • 3 * 4


    So the question is really:
    The number of factors is n
    The smallest number with n factors is X
    How do you find X?

    Please feel free to use an example because I think doing it all in algebra will be unreasonably difficult
    Last edited by Tzarian; April 26th 2012 at 12:52 PM.
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  2. #2
    Junior Member ignite's Avatar
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    Re: Smallest number with given number of factors

    Let N be any natural number and N={p_1}^{a_1}{p_2}^{a_2}...{p_n}^{a_n} , where p_1,p_2,...,p_n are prime numbers and a_1,a_2,a_n are natural numbers.
    Then number of factors of N is given by (a_1+1)(a_2+1)...(a_n+1)

    You are given number of factors say k.First step would be to factorize k.That is write k as (a_1+1)(a_2+1)...(a_n+1).
    Once you have found out a_1,a_2,..,a_n such that a_1 \ge a_2 ...\ge a_n , your x would be  2^{a_1} 3^{a_2} 5^{a_3} ....

    Main thing here is that you can multiple possibilities for  a_i 's and you have to optimize for that.

    Example:Given k=12,find x.
    k=3*2*2 . Corresponding x =  2^{(3-1)} 3^{(2-1)} 5^{(2-1)} =60
    k=6*2 . Corresponding x =  2^{(6-1)} 3^{(2-1)} = 96
    k=4*3 . Corresponding x =  2^{(4-1)} 3^{(3-1)} = 72
    So minimum x would be 60.
    Last edited by ignite; April 26th 2012 at 03:00 PM.
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