The number (101)¹⁰⁰ $$-$$ 1 is divisible by

The expression $2^{4 n}-15 n-1$, where $n \in \mathbb{N}$ (the set of natural numbers) is divisible by

Let $$P(n) = {3^{2n + 1}} + {2^{n + 2}}$$ where $$n \in N$$. Then