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Homework Help: Solving a polynomial congruence?

  1. Dec 13, 2011 #1
    1. The problem statement, all variables and given/known data

    Find one solution (or prove no solutions exist) to the equation

    x100 = 3 mod 83, where "=" means "congruent to"

    2. Relevant equations

    Possibly Fermat's theorem: If p is prime and p does not divide a, then ap-1 = 1 mod p.

    3. The attempt at a solution

    83 is prime. I know that 101 is prime, and so ap-1 == x^100 if p == 101. However, not sure how to deal with the mod 83 then.

    Alternatively, using p == 83, then x82 = 1 mod 83, but this doesn't seem to help much either.

    Any suggestions?
  2. jcsd
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