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Number Theory Q re: Euler phi-function

  1. Apr 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Let p be prime. Show that p n, where n is a positive integer, iff [itex]\phi[/itex](np) = (p-1)[itex]\phi[/itex](n).

    2. Relevant equations

    Theorem 1: If p is prime, then [itex]\phi[/itex](p) = p-1. Conversely, if p is a positive integer with [itex]\phi[/itex](p) = p-1, then p is prime.

    Theorem 2: Let m and n be relatively prime positive numbers. Then [itex]\phi[/itex](mn) = [itex]\phi[/itex](m)[itex]\phi[/itex](n).

    3. The attempt at a solution

    Assume that p n.
    By Theorem 2, [itex]\phi[/itex](np) = [itex]\phi[/itex](n)[itex]\phi[/itex](p).
    By Theorem 1, [itex]\phi[/itex](np) = [itex]\phi[/itex](n)[itex]\cdot[/itex](p-1).
    So, if p n, then [itex]\phi[/itex](np) = (p-1)[itex]\cdot[/itex][itex]\phi[/itex](n).

    Now, assume that [itex]\phi[/itex](np) = (p-1)[itex]\phi[/itex](n).
    By Theorem 1, [itex]\phi[/itex](np) = [itex]\phi[/itex](p)[itex]\phi[/itex](n), since p is prime.

    ...and, that's it. I can't use Theorem 2 to show that n and p are relatively prime, so I'm not entirely sure what to do next. p is prime, so n p, but I don't see anything to base a conclusion of p n.

    Any help would be greatly appreciated!
     
  2. jcsd
  3. Apr 15, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If a prime p doesn't divide n then aren't p and n relatively prime? A common divisor of p and n must divide p. Think about it.
     
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