Prove: if n is a natural number, then 2^2n − 1 must be divisible by 3.

Let p(n) be the statement. note 2^2 -1 = 3, divisible by 3hence p(1) is true.assume p(k) is true, that is, there exists integer q such that 2^(2k) -1 =3qthen 2^[2(k+1)] -1=[2^(2k)](4) -1=(3q+1)(4)-1=12q+3=3(4q+1)since (4q+1) is an integer, 2^[2(k+1)] -1 is divisible by 3, that is, p(k+1) is true.by the principle of mathematical induction, p(n) is true.