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Thread: Multiplicative Question

  1. #1
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    Multiplicative Question

    Assume$\displaystyle \sigma$ is multiplicative

    a) Let p be a prime and a $\displaystyle \in Z^+$. Prove that for an odd prime p, $\displaystyle \sigma(p^a)$ is odd if, and only f, a is even. Also prove that for p = 2, $\displaystyle \sigma(2^a)$ is odd for all a $\displaystyle \in Z^+.$

    b) Let n $\displaystyle \in Z^+$. Prove that $\displaystyle \sigma(n)$ is odd if, and only if, either n is a perfect square or twice a perfect square (i.e. $\displaystyle n \in {(m^2 \mid m \in Z^+)} \cup {(2m^2 \mid m \in Z})$
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    $\displaystyle \sigma(5^2)=6\cdot6=36$ (even)
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  3. #3
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    Quote Originally Posted by Also sprach Zarathustra View Post
    $\displaystyle \sigma(5^2)=6\cdot6=36$ (even)
    Wait, I have to prove that if p is an odd prime, then $\displaystyle \sigma(p^a)$ is odd when a is even. But $\displaystyle \sigma(5^2)$ is even which does not follow....
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