Apr 27 07:00 AM PDT  09:00 AM PDT Attend this webinar and master GMAT SC in 10 days by learning how meaning and logic can help you tackle 700+ level SC questions with ease. Apr 28 07:00 AM PDT  09:00 AM PDT Attend this webinar to learn a structured approach to solve 700+ Number Properties question in less than 2 minutes. Apr 29 08:00 AM PDT  09:00 AM PDT Join a free live webinar and learn timemanagement tactics that will guarantee you answer all questions, in all sections, on time. Save your spot today! Apr 30 10:00 PM PDT  11:00 PM PDT Enter to win 3 full months of access to EMPOWERgmat's groundbreaking GMAT prep course. Prize includes all 6 GMAT Official Practice exams and access to the GMAT Club Test & Quiz Bank Pack. May 01 10:00 PM PDT  11:00 PM PDT Target Test Prep is kicking off spring with a fresh giveaway contest! For a limited time, you have a chance to win 4 months of full, FREE access to our 5star rated GMAT Quant course.
Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 54544

k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
22 Sep 2015, 02:48
Question Stats:
77% (01:49) correct 23% (02:09) wrong based on 224 sessions
HideShow timer Statistics
k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9? A. 2^n  8 B. 2^n  2 C. 2^n D. 2^n + 4 E. 2^n + 5 Kudos for a correct solution.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________




Manager
Joined: 29 Jul 2015
Posts: 157

Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
22 Sep 2015, 04:18
Bunuel wrote: k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?
A. 2^n  8 B. 2^n  2 C. 2^n D. 2^n + 4 E. 2^n + 5
Kudos for a correct solution. Since 2^n + 7 is divisible by 9, 2^n+7+9 and 2^n+79 should also be divisible by 9. So our numbers are 2^n+16 and 2^n2 Answer: B




CEO
Joined: 12 Sep 2015
Posts: 3595
Location: Canada

Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
23 Sep 2015, 15:58
Bunuel wrote: k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?
A. 2^n  8 B. 2^n  2 C. 2^n D. 2^n + 4 E. 2^n + 5
Kudos for a correct solution. Kunal is using a nice rule that goes something like this: Given: k, M and N are integers If k is a divisor of both N and M, then k is a divisor of N+M (and N–M and M–N) We're told that 9 is a divisor of 2^n + 7 We also know that 9 is a divisor of 9. So, applying the above rule, 9 is a divisor of 2^n + 7 + 9, and 9 is a divisor of 2^n + 7  9For more on this rule and other divisor rules, see our free video: http://www.gmatprepnow.com/module/gmat ... /video/831Cheers, Brent
_________________
Test confidently with gmatprepnow.com



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 13981
Location: United States (CA)

Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
29 Sep 2015, 20:18
Hi All, This question can be solved with a bit of arithmetic and some 'brute force' (it also helps if you know the 'rule of 9' so that you can avoid having to do lots of division). We're given a series of facts to work with: 1) K = (2^N) + 7 where N is an integer that is greater than 1. 2) K is divisible by 9 We're asked which of the 5 answer choices MUST also be divisible by 9. ** Note: The 'rule of 9' can help you to quickly determine whether a number is divisible by 9 or not. To use this rule, you must add up the DIGITS of the number. If THAT total is divisible by 9, then the original number is divisible by 9. For example: 18 is divisible by 9 because the digits "1" + "8" = 9, which is divisible by 9 > thus, 18 IS divisible by 9 42 is NOT divisible by 9 because the digits "4" + "2" = 6, which is NOT divisible by 9 > thus, 42 is NOT divisible by 9 ** If you don't see the 'pattern' involved in this question, then that's fine  you can still get to the solution using basic arithmetic. We just have to figure out the smallest value of N that fits everything that we've been told. Let's start with... N = 2, so K = 4+7 = 11. 11 is NOT divisible by 9 though, so N CANNOT be 2 N = 3, so K = 15 > NOT divisible by 9, so N CANNOT be 3 N = 4, so K = 23 > NOT > N CANNOT be 4 N = 5, so K = 39 > NOT > N CANNOT be 5 N = 6, so K = 71 > NOT > N CANNOT be 6 N = 7, so K = 135. 135 IS divisible by 9...so N CAN be 7 Let's take THAT value of N and use it against the answer choices. When you plug N=7 into the 5 choices, only one of them is divisible by 9... Final Answer: GMAT assassins aren't born, they're made, Rich
_________________
760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****
Rich Cohen
CoFounder & GMAT Assassin
Special Offer: Save $75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/



DS Forum Moderator
Joined: 22 Aug 2013
Posts: 1443
Location: India

Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
11 Jun 2017, 23:59
The question asks which of the following MUST be divisible by 9. So if we take any value of n that satisfies the question stem, it should also satisfy the correct answer option.
Now k = 2^n + 7, is divisible by 9. We can see if we put n=1, k = 2^1+7 = 9, which is divisible by 9. So if we put n=1, it satisfies the given condition. We have to now put n=1 in the answer options and see which one satisfies:
A) 2^1  8 = 6, NOT divisible by 9 B) 2^1  2 = 0, YES. This is divisible by 9 C) 2^1 = 2, NOT divisible by 9 D) 2^1 + 4 = 6, NOT divisible by 9 E) 2^1 + 5 = 7, NOT divisible by 9
Hence B answer



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 13981
Location: United States (CA)

Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
12 Jun 2017, 11:51
amanvermagmat wrote: The question asks which of the following MUST be divisible by 9. So if we take any value of n that satisfies the question stem, it should also satisfy the correct answer option.
Now k = 2^n + 7, is divisible by 9. We can see if we put n=1, k = 2^1+7 = 9, which is divisible by 9. So if we put n=1, it satisfies the given condition. We have to now put n=1 in the answer options and see which one satisfies:
A) 2^1  8 = 6, NOT divisible by 9 B) 2^1  2 = 0, YES. This is divisible by 9 C) 2^1 = 2, NOT divisible by 9 D) 2^1 + 4 = 6, NOT divisible by 9 E) 2^1 + 5 = 7, NOT divisible by 9
Hence B answer Hi amanvermagmat, While you did get the correct answer, it's worth noting that you did NOT follow the instructions in the prompt. We're told that N is an integer GREATER than 1, so using N=1 was technically NOT appropriate. By extension, in this question, you were fortunate to get the correct answer. On Test Day, GMAT question writers will often design at least one wrong answer that will 'punish' someone who is not following the 'rules' of the prompt (meaning that if you don't implement some essential piece of information, then you will end up thinking that one of the wrong answers is correct; this is especially true of DS questions). As such, you should be careful about how you approach questions in the future (and on Test Day). GMAT assassins aren't born, they're made, Rich
_________________
760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****
Rich Cohen
CoFounder & GMAT Assassin
Special Offer: Save $75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/



DS Forum Moderator
Joined: 22 Aug 2013
Posts: 1443
Location: India

Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
12 Jun 2017, 20:59
EMPOWERgmatRichC wrote: amanvermagmat wrote: The question asks which of the following MUST be divisible by 9. So if we take any value of n that satisfies the question stem, it should also satisfy the correct answer option.
Now k = 2^n + 7, is divisible by 9. We can see if we put n=1, k = 2^1+7 = 9, which is divisible by 9. So if we put n=1, it satisfies the given condition. We have to now put n=1 in the answer options and see which one satisfies:
A) 2^1  8 = 6, NOT divisible by 9 B) 2^1  2 = 0, YES. This is divisible by 9 C) 2^1 = 2, NOT divisible by 9 D) 2^1 + 4 = 6, NOT divisible by 9 E) 2^1 + 5 = 7, NOT divisible by 9
Hence B answer Hi amanvermagmat, While you did get the correct answer, it's worth noting that you did NOT follow the instructions in the prompt. We're told that N is an integer GREATER than 1, so using N=1 was technically NOT appropriate. By extension, in this question, you were fortunate to get the correct answer. On Test Day, GMAT question writers will often design at least one wrong answer that will 'punish' someone who is not following the 'rules' of the prompt (meaning that if you don't implement some essential piece of information, then you will end up thinking that one of the wrong answers is correct; this is especially true of DS questions). As such, you should be careful about how you approach questions in the future (and on Test Day). GMAT assassins aren't born, they're made, Rich Hi Yes, I missed it. Thank you for pointing that out. Kudos.



Intern
Joined: 21 Aug 2015
Posts: 22

Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
13 Jun 2017, 01:44
Another approach can be
We know that k = 2^n + 7 is divisible by 9 so we can say that 2^n when divided by 9 is leaving a remainder 2 so definitely 2^n  2 must be divisible by 9.
Please let me know if my approach is right.



Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2825

Re: k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
06 Jul 2017, 17:16
Bunuel wrote: k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?
A. 2^n  8 B. 2^n  2 C. 2^n D. 2^n + 4 E. 2^n + 5 Since k/9 = integer, (2^n + 7)/9 = integer. Since 2^n + 7 is a multiple of 9, we can add or subtract any multiple of 9 to 2^n + 7 and that expression will still be divisible by 9. Thus: 2^n + 7  9 = 2^n  2 MUST be divisible by 9. Answer: B
_________________
5star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews



Senior SC Moderator
Joined: 22 May 2016
Posts: 2654

k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
Show Tags
07 Jul 2017, 14:23
Bunuel wrote: k = 2^n + 7, where n is an integer greater than 1. If k is divisible by 9, which of the following MUST be divisible by 9?
A. 2^n  8 B. 2^n  2 C. 2^n D. 2^n + 4 E. 2^n + 5
Kudos for a correct solution. I used brute force. Powers of 2 are easy to list, and all we need is a number + 7 that is divisible by 9. If number gets large, all we need is a number whose digits sum to 9. Powers of 2 (must be greater than 1): 2\(^2\), 2\(^3\), 2\(^4\), 2\(^5\), 2\(^6\), 2\(^7\), 2\(^8\) = 4, 8, 16, 32, 64, 128, 256 As I wrote each power down, I added 7 to see if the result would be divisible by 9, and hence would be k. I did the divisibility part in my head until I reached 128 + 7 = 135. But digits of 135 sum to 9; 135 is divisible by 9. So 135 = 128 + 7, and 128 = 2\(^7\), n = 7 Then I reasoned backward. All the answer choices have 2\(^n\). n = 7. Starting point, therefore, for all answers is 128. If 128 must have 7 added to it to be divisible by 9 ... Go the other way on the number line, from 128, to reach another multiple of 9. If 7 up from a number equals multiple of 9, then 2 down from a number will also be a multiple of 9. "2 down" from 128 = 128  2. 128  2 = 126. Those digits also sum to 9, and per prompt, 126 must be divisible by 9. 128  2 = 2\(^n\)  2. Answer B
_________________
Listen, are you breathing just a little, and calling it a life?  Mary Oliver
For practice SC questions with official explanations that were posted and moderated by the SC Team, go to SC Butler here: https://gmatclub.com/forum/projectscbutlerget2scquestionseveryday281043.html




k = 2^n + 7, where n is an integer greater than 1. If k is divisible
[#permalink]
07 Jul 2017, 14:23






