1. ## divisibility proof

How would I go about proving 165 | (n^20 - a^20) if n and a are relatively prime to 165?

2. First: $165=3\cdot{5}\cdot{11}$

By Fermat's Little Theorem $n^{4}\equiv{1}(\bmod.5)$ (since n is coprime to 5) then $n^{20}\equiv{1^5=1}(\bmod.5)$ and in the same way: $a^{20}\equiv{1}(\bmod.5)$

THus: $n^{20}-a^{20}\equiv{0}(\bmod.5)$ so 5 divides $n^{20}-a^{20}$

Do the same for the other two primes ( 3 and 11) and you have that 165 divides $n^{20}-a^{20}$