If n is a positive integer, prove that `3^(3n) - 26n-1` is divisible by 676

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

If n is any positive integer , show that 2^(3n +3) -7n - 8 is divisible by 49 .

For every positive integer n, prove that 7^n-3^n is divisible by 4.

Prove by induction that if n is a positive integer not divisible by 3 , then 3^(2n)+3^(n)+1 is divisible by 13 .

If p is a natural number, then prove that p^(n+1) + (p+1)^(2n-1) is divisible by p^(2) + p +1 for every positive integer n.