For Statement (1) if n = 1, then 2^n = 2 and the remainder of 2 / 3 is 2 ==>statement I may NOT be true.
For statement (II) if 3^n= (-3)^n, then n must be even i.e. n = 0, 2, 4, 6, 8, …. When n = 0, 2 ^ n = 1 and the remainder of 1/ 3 is 1. when n is other even numbers, 2^n is 4, 16, 64, 256, …=4^m.
Note that the remainder of (4^m /3 )must be 1. The reason is as follows:
4*4=(3+1)(3+1)=3*3+3*1+1*3+1*1==>The blue portion is a multiple of 3==>3p+1
so,,4^2*4=(3p+1)*4=(3p+1)(3+1)==>Repeat the exercise as above ==>3r+1==>we can write 4^m =3x+1 for any m, x belongs to Integer.
so, if 4^m is divided by 3, the remainder must be 1 ==> statement II must be true.
For statement III. Sqrt(2^n ) belongs to integers ==> n must be even. ==> statement III must be true. (The reason is the same as given above)
OA : E