<s>[INST] 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. 
 [/INST] Since $n$ is even:
$20^n + 16^n - 3^n - 1 \equiv 1^n + (-3)^n - 3^n - 1 = 1 + 3^n - 3^n - 1 = 0 \bmod 19$
Also:
$20^n + 16^n - 3^n - 1 \equiv 3^n + (-1)^n - 3^n - 1 = 3^n + 1 - 3^n - 1 = 0 \bmod 17$
So the quantity is divisible by $17\times 19 = 323$
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