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When is x^n congruent to x^m (mod 3) for all x in Z^+?

  1. Sep 7, 2016 #1
    (mentor note: moved here from another forum hence no template)

    Hello, I need some help. For which ##m,n \in \mathbb{Z^+}## is ##x^n \equiv x^m \space \text{(mod 3)}## for all ##x \in \mathbb{Z^+}##? I have no clue how to solve this. According to the answer, ##n - m ## is an even integer. Anyone who can point me in the right direction? Thanks.
    Last edited by a moderator: Sep 7, 2016
  2. jcsd
  3. Sep 7, 2016 #2


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    Staff: Mentor

    There are just three relevant cases for x, check them individually and you should get the right answer.
  4. Sep 7, 2016 #3


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    Homework Helper

    A quick simplification is to note:
    ##x^m \mod 3 \equiv (x \mod 3)^m \mod 3. ##
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