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.

n^(3)-n is divisible by 6, for each natural numbers n ge2

Prove the statement by the principle of mathematical induction : 4^n - 1 is divisible by 3, for each natural number n .

Use induction to prove that n^(3) - n + 3 , is divisible by 3, for all natural numbers n

Prove that 6^(2n)-35n-1 is divisible by 1225 for all natural numbers of n.