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?