For a non-negative integer n,

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jack0997: Junior | Next Rank: 30 Posts; Posts: 18; Joined: Mon Nov 11, 2013 5:31 am; Thanked: 1 times

by jack0997 » Tue Jun 13, 2017 1:20 am

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 = Âƒ 3^(-n)
III. âˆš(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

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Jun 13, 2017 1:52 am

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 = 3^(-n)
III. âˆš(2^n) is an integer.

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

Try to show that I, II and III do NOT have to be true.

Case 1: n=0, with the result that (2^n)/3 = 2â�°/3 = 1/3 = 0 R1
If n=0, option I is not true.
Eliminate any answer choice that includes I.
Eliminate A and D.

Case 2: n=2, with the result that (2^n)/3 = 2Â²/3 = 4/3 = 1 R1
If n=2, option II is not true.
Eliminate any remaining answer choice that includes II.
Eliminate B and E.

The correct answer is C.

Last edited by GMATGuruNY on Tue Jun 13, 2017 2:07 am, edited 1 time in total.

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Tue Jun 13, 2017 2:01 am

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 = Âƒ 3^(-n)
III. âˆš(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 = Âƒ 3^(-n): 3^n = Âƒ 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. âˆš(2^n) is an integer: MUST-BE true! Since âˆš(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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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Sun Jun 18, 2017 10:12 am

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.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Thu Jun 22, 2017 9:41 pm

Look for the pattern too.

2â�°, 2Â², 2â�´, ... all have remainder 1

2Â¹, 2Â³, 2â�µ, ... all have remainder 2

So we need to have an even exponent. III follows from there.

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