jack0997 wrote:For a non-negative integer n, if the remainder is 1 when 2^n is divided by 3, then which of the following must be true?

I. n > 0
II. 3^n = ƒ 3^(-n)
III. √(2^n) is an integer.

(A) Only I
(B) Only II
(C) Only III
(D) Only I and III
(E) Only II and III

OA C

2^n divided by 3 returns a remainder '1,' if n = 0, 2, 4, even number.

1. 2^0 = 1; 1 divided by 3 retuns 1 as remiainder;
2. 2^1 = 2; 2 divided by 3 retuns 2 as remiainder, not 1
3. 2^2 = 4; 4 divided by 3 retuns 1 as remiainder
4. 2^3 = 8; 8 divided by 3 retuns 2 as remiainder, not 1
5. 2^4 = 16; 16 divided by 3 retuns 1 as remiainder

Let's analyze each statement one by one.

I. n > 0: Incorrect. It's correct if n is even, else incorrect
II. 3^n = ƒ 3^(-n): 3^n = ƒ 3^(-n) => n = 0; however, it is not a MUST-BE true statement. If n = 2,4,6, or an even number, 2^n divided by 3 also returns 1 as remainder.
III. √(2^n) is an integer: MUST-BE true! Since √(2^n) is an integer, n must be EVEN.

The correct answer: C

Hope this helps!

Relevant book: Manhattan Review GMAT Sets & Statistics Guide

-Jay
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Here's a general tip: when you see the phrase "non-negative," think about zero! Many people automatically interpret "non-negative" as "positive." But when the GMAT means "positive," they'll say "positive"! If they say "non-negative," it means that the 0 possibility is relevant.

Always start by plugging in 0 in that case.