Prove that `(n !+1)` is not divisible by any natural number between `2` and `n`

Prove that (n !+1) is not divisible by any natural number between 2a n dndot

Prove that n!+1 is not divisible by any integer between 2 and n.

prove that 4^(n)-1 is divisible by 3, for each natural number n.

prove that 4^(n)-1 is divisible by 3, for each natural number n.

prove that 3^(2n)-1 is divisible by 8, for all natural numbers n.

prove that 3^(2n)-1 is divisible by 8, for all natural numbers n.

prove that n(n^(2)+5) is divisible by 6, for each natural number n.

prove that n(n^(2)+5) is divisible by 6, for each natural number n.

Using principle of mathematical induction, prove that 7^(4^(n)) -1 is divisible by 2^(2n+3) for any natural number n.

Using principle of mathematical induction, prove that 7^(4^(n)) -1 is divisible by 2^(2n+3) for any natural number n.