Prove if n is a natural number then 22n 1 must be divisible

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

Solution

lets prove by induction.
for n=1:

2^(2n)-1=4-1=3

so it is divisible by 3.

now assume that its true for n=k

then 2^(2k)-1=3*k1 where k1 is an integer

now for n=k+1

2^(2k+2)-1=2^(2k)*4-1=4*(3*k1+1)-1=12*k1+3

=3*(4k1+1)

hence it is also divisilbe by 3.

hence proved.