1. The problem statement, all variables and given/known data x cong 1(mod m^k) implies x^m cong 1(mod m^(k+1)) 2. Relevant equations x cong 1(mod m^k) <=> m^k|x-1 <=> ym^k=x-1 3. The attempt at a solution starting with ym^k=x-1 add one to both sides ym^k+1=x now rise to the power m. (ym^k+1)^m=x^m <=> subtract the 1^m from the end of the expantion to get x^m-1^m-(...)=(y^m)(m^(km)) where (...) is the rest of the binomial expansion. I am stuck here. somehow I need to get a m^(k+1) on the LHS and say it divides everything on the RHS. I don't think my approach is the best.