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Math Help - Arithmetic Functions.

  1. #1
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    Arithmetic Functions.

    a) Let n be a positive integer. Prove that phi(n)=(n/2) if and only if n=2^k. for some positive integer k.

    b) Let n be a positive integer. Suppose that the product of all of the positive divisors of n is n^2. prove that v(n)=4. (v is nu, the product of all positive divisors)
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  2. #2
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    Quote Originally Posted by JCIR View Post
    a) Let n be a positive integer. Prove that phi(n)=(n/2) if and only if n=2^k. for some positive integer k.
    Hint: Just write n=2^a p_1^{a_1} ... p_j^{a_j}.
    And use the formula.

    b) Let n be a positive integer. Suppose that the product of all of the positive divisors of n is n^2. prove that v(n)=4. (v is nu, the product of all positive divisors)
    Hint: The product of the positive divisors is n^{\tau(n)/2}.
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