$7^{2n} + 16n - 1$ $(n \in N)$ is divisible by

For every natural number $n,$ $n(n^2 - 1)$ is divisible by

Among the statements:
$(S1):$ $2023^{2022} - 1999^{2022}$ is divisible by $8$.
$(S2):$ $13(13^{n}) - 11n - 13$ is divisible by $144$ for infinitely many $n \in N$.

For all positive integers $n$,if $3(5^{2n+1}) + 2^{3n+1}$ is divisible by $k$,then the number of prime numbers less than or equal to $k$ is:

Let $n \in \mathbb{N}$. Which one of the following is true?

The expression ${49^n} + 16n - 1$ is divisible by: