For every positive integer n, prove that `7^(n) – 3^(n)` is divisible by 4.

For any positive integer n, prove that (n^(3) - n) is divisible by 6.

n^7-n is divisible by 42 .

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

Prove that 2.7^(n)+ 3.5^(n)-5 is divisible by 24 for all n in N

3^(2n+2)-8n-9 is divisible by 8.

Prove that (n !) is divisible by (n !)^(n-1)!

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.

Prove by induction that n(n+1) (2n+1) is divisible by 6.

10^n+3(4^(n+2))+5 is divisible by (n in N)