For a non-negative integer n,

This topic has expert replies
Junior | Next Rank: 30 Posts
Posts: 18
Joined: Mon Nov 11, 2013 5:31 am
Thanked: 1 times

For a non-negative integer n,

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 = 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

User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
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. √(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 = 2�/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 = 2²/3 = 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.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 3008
Joined: Mon Aug 22, 2016 6:19 am
Location: Grand Central / New York
Thanked: 470 times
Followed by:34 members

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 = 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
_________________
Manhattan Review GMAT Prep

Locations: New York | Mumbai | Ho Chi Minh City | Budapest | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 2095
Joined: Tue Dec 04, 2012 3:22 pm
Thanked: 1443 times
Followed by:247 members

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

GMAT Instructor
Posts: 2630
Joined: Wed Sep 12, 2012 3:32 pm
Location: East Bay all the way
Thanked: 625 times
Followed by:119 members
GMAT Score:780

by Matt@VeritasPrep » Thu Jun 22, 2017 9:41 pm
Look for the pattern too.

2�, 2², 2�, ... all have remainder 1

2¹, 2³, 2�, ... all have remainder 2

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