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Math Help - tau(n) proof

  1. #1
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    tau(n) proof

    How would I find the smallest number n such that tau(n)=1001

    tau(n) is the number of possible divisors of a positive number, n
    Last edited by MichaelG; December 15th 2009 at 12:36 PM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    What is tau?

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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Given  n=p_1^{\alpha_1}\cdot\cdot\cdot p_k^{\alpha_k} ,

     \tau(n) = \prod_{i=1}^{k} (\alpha_i+1) .

     1001 = 7\cdot 11\cdot 13

    Therefore  \alpha_1 = 6 ,  \alpha_2 = 10 ,  \alpha_3 = 12 .

    Now pair the smallest prime with the largest  \alpha , the second smallest prime with the second larget  \alpha , and the third smallest prime with the smallest  \alpha .

    Hence  n=2^{12}\cdot 3^{10}\cdot 5^6=3779136000000 .
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