1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook