How would I go about proving 165 | (n^20 - a^20) if n and a are relatively prime to 165?
First: $\displaystyle 165=3\cdot{5}\cdot{11}$
By Fermat's Little Theorem $\displaystyle n^{4}\equiv{1}(\bmod.5)$ (since n is coprime to 5) then $\displaystyle n^{20}\equiv{1^5=1}(\bmod.5)$ and in the same way: $\displaystyle a^{20}\equiv{1}(\bmod.5)$
THus: $\displaystyle n^{20}-a^{20}\equiv{0}(\bmod.5)$ so 5 divides $\displaystyle n^{20}-a^{20}$
Do the same for the other two primes ( 3 and 11) and you have that 165 divides $\displaystyle n^{20}-a^{20}$