Assume $n$ is even. Prove that $323$ divides $20^n+16^n-3^n-1$. I'm unclear what is the best method to teach this with minimum math experience. 

We have $323 = 17 \cdot 19$ with $\gcd(17, 19) = 1$. Also, $19 = 20-1 \mid 20^n - 1, 19 = 16 + 3 \mid 16^n - 3^n \Rightarrow 19 \mid 20^n+16^n-3^n-1$
and $17 = 20 - 3 \mid 20^n - 3^n$, $17 = 16+1 \mid 16^n +1 \Rightarrow 17 \mid 20^n+16^n-3^n-1$
$\gcd(17,19)=1 \Rightarrow 323 = 17 \cdot 19 \mid 20^n+16^n-3^n-1$
