Product of any r consecutive natural numbers is always divisible by

For each n $$\in$$ N, 2³ⁿ $$-$$ 1 is divisible by

here N is a set of natural numbers.

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