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Fermat's/Euler problem

  1. Apr 5, 2010 #1
    I've got a pair of (related) problems that are keeping me stumped. The two problems they are asking me to prove are:

    [tex] p^{(q-1)} + q^{(p - 1)} \equiv 1 \; (\!\!\!\!\!\! \mod \, pq) [/tex]

    [tex]a^{\phi(b)} + b^{\phi(a)} \equiv 1
    \; (\!\!\!\!\!\! \mod \, ab)[/tex]

    Where [tex] \phi(n) [/tex] is Euler's Totient Function.

    I know that these are similar, as the second problem is using Euler's Theorem, a generalization of Fermat's Little Theorem, I just can't seem to figure them out.
  2. jcsd
  3. Apr 5, 2010 #2
    for the first one, see if this helps...

    By Fermat's Theorem (assuming that p and p are relatively prime...) we have
    [tex]q^p=q [/tex]mod [tex]p [/tex]. We can cancel q from both sides since gcd(q,p) = 1, so [tex]q^{(p-1)} = 1 (mod p)[/tex]. Also [tex] p^{(q-1)} = 0 (mod p) [/tex] we get

    [tex]p^{(q-1)} + q^{(p-1)} = 1 (mod p) [/tex]

    can you go from there?
    Last edited: Apr 5, 2010
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