Non-negative integer

[bhumika.k.shah]

For a nonnegative integer n, if the remainder is 1 when 2^n is divided by 3, then which of
the following must be true?

I. n is greater than zero.
II. 3^n = (-3)^n
III. √2^n is an integer.

A. I only
B. II only
C. I and II
D. I and III
E. II and III

[ace_gre]

IMO E

First determine the pattern 2 ^0=1, remainder when divided by 3=1
2^1=2, remainder when divided by 3=2
2^2=4, remainder=1
2^3=8 , remainder = 2
2^4=16 ,remainder =1
2^5=32 , remainder =2

Alternating pattern of 1 and 2. In all cases remainder is 1 when n=even.

I is not true. For n=0, remainder =1.
II For n=even, 3^n=(-3)^n. True
III √2 ^ n= integer(for n=0,2,4,..value is integer). True

Hence II and III must be true.

[mehravikas]

Agree answer should be D