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Math Help - how should I solve this question? number of divisors

  1. #1
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    how should I solve this question? number of divisors

    Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including itself and 1. Find n/75.

    Thank you very much.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Jenny20 View Post
    Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including itself and 1. Find n/75.

    Thank you very much.
    Suppose the required number is N, and that it has prime decomposition:

    N=p_1^(n_1) p_2^(n_2), ... , p_m^(n_m).

    where n_1, n_2 ..are all >=1. Then the number of divisors this number has (including 1 and N) is:

    d=(n_1+1)(n_2+1)...(n_m+1).

    Now if d=75, as 75=3.5^2, any factorisation of d has at most 3 factors (they
    are 3, 5, 5). Also we already know that two of the p_i 's are 3 and 5, the
    smallest N with the required properties has 2 as a factor with a multiplicity
    as large as can be made consistent with the conditions on N, so:

    Making the correction Plato pointed out:

    N=2^4 3^4 5^2 = 32400.

    RonL
    Last edited by CaptainBlack; December 25th 2006 at 09:41 AM. Reason: correction to final result
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  3. #3
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    I have found a smaller solution than the one above.
    In fact N=2^5 3^5 5^3 has (6)(6)(4)=144 positive integral factors.
    If we take N=(2^4)(3^4)(5^2)=32400 that has 75 positive integral factors.
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  4. #4
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    Hi plato,

    I also think n should be equal to 32400 because it has exactly 75 positive integral divisors.

    Could you please show me how to get this number? Thank you very much.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Plato View Post
    I have found a smaller solution than the one above.
    In fact N=2^5 3^5 5^3 has (6)(6)(4)=144 positive integral factors.
    If we take N=(2^4)(3^4)(5^2)=32400 that has 75 positive integral factors.
    Mistake on my part the number of factors has +1's in it that seems to have dropped out of my calculations at some point

    So if I had done my sums in a manner consistent with what I had said I
    would have had tha same answer.

    RonL
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  6. #6
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    Quote Originally Posted by Jenny20 View Post
    Could you please show me how to get this number?
    Read CaptainBlacks’ explication; its logic is correct, but he just overdid the exponents. If (2^a)(3^b)(5^c) is to have 75 positive integral factors, each of a,b, & c is great than or equal to 0 and (a+1)(b+1)(c+1)=75. That means c is at least 2 to get 25 and b is at least 1 to get 75.
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