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

Prove that 3^(2n)+24n-1 is divisible by 32 .

n^2-1 is divisible by 8, if n is

Prove that 33! Is divisible by 2^(19) and what is the largest integer n such that 33! Is divisible by 2^(n) ?

Use the Principle of Mathematical Induction to prove that n(n + 1) (2n + 1) is divisible by 6 for all n in N.

Show that 9^(n+1) -8n -9 is divisible by 64, whenever n is a positive integer.