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Math Help - I need a help on a couple problems...

  1. #1
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    I need a help on a couple problems...

    I am having trouble with 2 problems for my class. They are:

    1. Find all values of n such that -12=22(mod n).

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

    Thanks for the help!
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  2. #2
    Super Member PaulRS's Avatar
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    1. -12\equiv{22}(\bmod.n)

    Now sum 12 on both sides: 0\equiv{34}(\bmod.n)

    That means that n must divide 34, so find the divisors of 34.

    2. Remember that: \phi(n)=n\cdot{\prod_{p|n}{\left(1-\frac{1}{p}\right)}}
    (where by p I mean prime)

    If: \phi(n)=\frac{n}{2} we have: \frac{1}{2}=\prod_{p|n}{\left(1-\frac{1}{p}\right)}

    And that can happen iff 2 is the only prime divisor of n. (Try working with divisibility)
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