1. The problem statement, all variables and given/known data Number theory problem. we are just doing modular division and congruence theory. x^5==x(mod10) ==> 10|x^5-x 2. Relevant equations induction. let x=1,x=x+1 3. The attempt at a solution let x=1 1^5==1(mod10) this is trivial but 2^5==2(mod10)... let x=x+1 (x+1)^5==(x+1)mod10 ==> 10|(x+1)^5-(x+1) (x+1)^5=x^5+(....)+1 ==> 10|x^5-x+(...) now we know that 10|x^5-x by assumption. so that part is done just need to show that 10|(...) (...)=5x^4+10x^3+10x^2+5x obviously 10|10x^3+10x^2 so we require 10|5x^4+5x. 10|5x(x^3+1). if x is even than we can factor out a 2 and get 10|10x/2(x^3+1) and be done. if x is odd. then x^3+1 is always even and we can factor out a 2 and get 10|10x(x^3+1)/2. does this last part need proving? can you give me a better way of proving this? induction often works, but i like finding clever ways of doing proofs instead of brute force with induction.