Binomial theroem problem

Question

I have a problem and i think that will be solved by binomial theorem I am not sure:

$20^n+16^n-3^n-1$ get $n$ to make that expression divisible by $323$.

Answer 1

As $323=17\times 19$ we just need divisibility by both $17$ and $19$.

Working $\pmod {17}$: our expression becomes $$3^n+(-1)^n-3^n-1=(-1)^n-1$$

Thus, to get divisibility by $17$ we need $n$ to be even.

Working $\pmod {19}$: our expression becomes $$1^n+(-3)^n-3^n-1=(-3)^n-3^n$$

Thus, to get divisibility by $19$ we need $n$ to be even.

We conclude that any even $n$ will work.

Answer 2

So we look for a number that is a multiple of $\;17\cdot19\;$, so:

$$20^n+16^n-3^n-1=\begin{cases}3^n+(-1)^n-3^n-1\pmod{17}=0\pmod{17}\implies\;n\;\text{even}\\{}\\1^n+(-3)^n-3^n-1\pmod{19}=0\pmod{19}\implies\;n\;\text{even}\end{cases}$$

Thus, for example, with $\;n=2\;$ :

$$20^2+16^2-3^2-1=646=2\cdot343$$