How can I show that (a^n -b^n) doesn't divide (a^n + b ^n) for all integers a,b, and n? I have that if (a^n -b^n) did divide (a^n + b ^n), then (a^n +b^n) = q (a^n -b^n) which implies b^n = -q*b^n (mod a^n). Then 1 = -q (mod a^n), meaning gcd(b, a^n) = 1. I am unsure of what more I can deduce. Any help is appreciated.