Mar 23 07:00 AM PDT  09:00 AM PDT Christina scored 760 by having clear (ability) milestones and a trackable plan to achieve the same. Attend this webinar to learn how to build trackable milestones that leverage your strengths to help you get to your target GMAT score. Mar 27 03:00 PM PDT  04:00 PM PDT Join a free live webinar and learn the winning strategy for a 700+ score on GMAT & the perfect application. Save your spot today! Wednesday, March 27th at 3 pm PST Mar 29 10:00 PM PDT  11:00 PM PDT Right now, their GMAT prep, GRE prep, and MBA admissions consulting services are up to $1,100 off. GMAT (Save up to $261): SPRINGEXTRAGMAT GRE Prep (Save up to $149): SPRINGEXTRAGRE MBA (Save up to $1,240): SPRINGEXTRAMBA
Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 53771

Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 03:52
Question Stats:
76% (01:24) correct 24% (01:34) wrong based on 177 sessions
HideShow timer Statistics




Math Expert
Joined: 02 Aug 2009
Posts: 7425

Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 08:44
stonecold wrote: Here 2^(any power ≥ 11) is a factor of three. Hence using the rule => Multiple  multiple is always a multiple We must subtract a multiple of 3 I.e we must subtract 3 Smash that C for me. I hope i am not missing anything Hi, this is not correct.. \(2^{11} or 2^{254312}\)will have ONLY 2 as prime factor A simple way would be \(2^1 =2\) , which requires 1 to be added for it to be div by 3.. \(2^2 = 4\) requires 1 to be subtracted .. and so on.. basically an ODD power of 2 requires 1 to be added to the number to be div by 3.. and any EVEN power would require 1 to be subtracted..here 256 is EVEN, so it requires 1 to be subtracted so \(2^{256}  1\) is div by 3 ans A
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html 4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentageincreasedecreasewhatshouldbethedenominator287528.html
GMAT Expert




Current Student
Joined: 12 Aug 2015
Posts: 2616

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 06:15



Current Student
Joined: 20 Dec 2015
Posts: 1
Location: United States
GPA: 3.61

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 08:28
stonecold wrote: Here 2^(any power ≥ 11) is a factor of three. Hence using the rule => Multiple  multiple is always a multiple We must subtract a multiple of 3 I.e we must subtract 3 Smash that C for me. I hope i am not missing anything I may be missing something here, but are you certain that your power of 2 rule checks out? I tried 2^15 and others. Unless I'm missing something, I don't think this would give the correct answer.



CEO
Joined: 20 Mar 2014
Posts: 2624
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 08:46
stonecold wrote: Here 2^(any power ≥ 11) is a factor of three.Hence using the rule => Multiple  multiple is always a multiple We must subtract a multiple of 3 I.e we must subtract 3 Smash that C for me. I hope i am not missing anything Your statement above (in red) is NOT correct. Important point, 2^number where number >11 can not be a FACTOR of 3 but will be a MULTIPLE of 3. 2^12 is definitely NOT a multiple of 3 as 2^12 will only have 2s in it. Coming back to the question, 2^1 leaves a remainder of 2 when divided by 3 2^2 leaves a remainder of 1 when divided by 3 2^3 leaves a remainder of 2 when divided by 3 2^4 leaves a remainder of 1 when divided by 3... etc. and the cyclicity continues. Thus, \(2^{526}\) will leave a remainder of 1 when divided by 3. Thus you must subtract 1 from \(2^{526}\) to make it divisible by 3. A is thus the correct answer. Hope this helps.



Current Student
Joined: 12 Aug 2015
Posts: 2616

Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 08:49



Manager
Joined: 09 Oct 2015
Posts: 241

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 10:57
Anyway binomial expansion would help solve this? Basically upon opening the binomial expansion, everything but the last digit would be divisible by 3. (31)^n
Posted from my mobile device



Current Student
Joined: 12 Aug 2015
Posts: 2616

Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 11:03
rahulkashyap wrote: Anyway binomial expansion would help solve this? Basically upon opening the binomial expansion, everything but the last digit would be divisible by 3. (31)^n
Posted from my mobile device Hey rahulkashyapYes I think we can use that here 2^526 = (31)^526 => expanding using (ab)^n => all terms will have 3 but the last Expanding the same => 3^526 * .........(1)^526 => 3p+1 for some integer p hence the remainder will be 1 P.S => Thank you for reminding me that Binomial can be a Saviour on the GMAT find the remainder Question..!!
_________________
Give me a hell yeah ...!!!!!
MBA Dating: BSCHOOL with the MOST ATTRACTIVE Women MBA Recruiting: EMPLOYMENT AND SALARY STATISTICS AT TOP BSCHOOLS IN THE US! (2018) MBA Financing: INDIAN PUBLIC BANKS vs PRODIGY FINANCE!
Getting into HOLLYWOOD with an MBA! The MOST AFFORDABLE MBA programs! STONECOLD's BRUTAL Mock Tests for GMATQuant(700+) AVERAGE GRE Scores At The Top Business Schools!



CEO
Joined: 20 Mar 2014
Posts: 2624
Concentration: Finance, Strategy
GPA: 3.7
WE: Engineering (Aerospace and Defense)

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 11:21
stonecold wrote: rahulkashyap wrote: Anyway binomial expansion would help solve this? Basically upon opening the binomial expansion, everything but the last digit would be divisible by 3. (31)^n
Posted from my mobile device Hey rahulkashyapYes I think we can use that here 2^526 = (31)^526 => expanding using (ab)^n => all terms will have 3 but the last Expanding the same => 3^526 * .........(1)^526 => 3p+1 for some integer p hence the remainder will be 1 P.S => Thank you for reminding me that Binomial can be a Saviour on the GMAT find the remainder Question..!! Most, if not all, remainder questions in GMAT can be solved by cyclicity. So, yes, use Binomial theorem if you are comfortable with it but do not forget about cyclicity (I did not use Binomial Theorem but used cyclicity instead).



Board of Directors
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4404
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 12:36
Bunuel wrote: Which of the following must be subtracted from 2^526 so that the resulting integer will be a multiple of 3?
A. 1 B. 2 C. 3 D. 5 E. 6 No need to go till the fourth power of 2 ; second power of 2 is sufficient\(2^{526}\) = \({2}^{2*128}\) \({2}^{2*128}\) = \(4^{128}\) 4/3 will produce 1 as remainder 1^128 = 1 Hence remainder will be 1 PS : Binomial Theorem is not in GMAT Syllabus , however if one is through with it , its his/her choice  Objective is to get the answer correct in whatever method you are confident with in least amount of time...Abhishek
_________________
Thanks and Regards
Abhishek....
PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS
How to use Search Function in GMAT Club  Rules for Posting in QA forum  Writing Mathematical Formulas Rules for Posting in VA forum  Request Expert's Reply ( VA Forum Only )



Current Student
Joined: 12 Aug 2015
Posts: 2616

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 12:45
Abhishek009 wrote: Bunuel wrote: Which of the following must be subtracted from 2^526 so that the resulting integer will be a multiple of 3?
A. 1 B. 2 C. 3 D. 5 E. 6 No need to go till the fourth power of 2 ; second power of 2 is sufficient\(2^{526}\) = \({2}^{2*128}\) \({2}^{2*128}\) = \(4^{128}\) 4/3 will produce 1 as remainder 1^128 = 1 Hence remainder will be 1 PS : Binomial Theorem is not in GMAT Syllabus , however if one is through with it , its his/her choice  Objective is to get the answer correct in whatever method you are confident with in least amount of time...AbhishekHey Just one doubt i have => when you say 4/3 gives one as the remainder ; you are essentially writing 4 as 3P+1 so (3p+1)^128 Isn't that binomial ?? Clearly that isnt cyclicity Regards Stonecold
_________________
Give me a hell yeah ...!!!!!
MBA Dating: BSCHOOL with the MOST ATTRACTIVE Women MBA Recruiting: EMPLOYMENT AND SALARY STATISTICS AT TOP BSCHOOLS IN THE US! (2018) MBA Financing: INDIAN PUBLIC BANKS vs PRODIGY FINANCE!
Getting into HOLLYWOOD with an MBA! The MOST AFFORDABLE MBA programs! STONECOLD's BRUTAL Mock Tests for GMATQuant(700+) AVERAGE GRE Scores At The Top Business Schools!



Board of Directors
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4404
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
25 Apr 2016, 13:01
stonecold wrote: Hey Just one doubt i have => when you say 4/3 gives one as the remainder ; you are essentially writing 4 as 3P+1 so (3p+1)^128 Isn't that binomial ?? Clearly that isnt cyclicity Regards Stonecold[/quote] {4^1} / 3 =4/3 remainder 1 {4^2} / 3 = 16/3 remainder 1 {4^3} / 3 = 64/3 remainder 1 {4^4} / 3 = 256/3 remainder 1 Actually the same remainder keeps repeating .....Try with a diff no, say 2 {2^1}/3 = remainder 2 {2^2}/3 = remainder 1 {2^3}/3 = remainder 2 {2^4}/3 = remainder 1 Se there is a cyclic pattern.... Hope this helps , we are using this logic to solve the problem and/ or any other problem involving remainder.
_________________
Thanks and Regards
Abhishek....
PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS
How to use Search Function in GMAT Club  Rules for Posting in QA forum  Writing Mathematical Formulas Rules for Posting in VA forum  Request Expert's Reply ( VA Forum Only )



Intern
Joined: 24 Mar 2016
Posts: 4

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
28 Apr 2016, 04:07
Bunuel wrote: Which of the following must be subtracted from 2^526 so that the resulting integer will be a multiple of 3?
A. 1 B. 2 C. 3 D. 5 E. 6 The Answer is A) as a lot of people have written before me. But what I want to add about this question, is that if you don't have an approach on how to solve it, it is very easy to see that only option A) and B) could be true. If it would be C, well then the resulting integer would already be a multiple of 3. If it would be D), then it would also be true for B) since 52=3. The same counts for E) 63=3 which is C).



Intern
Joined: 23 Apr 2016
Posts: 7

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
28 Apr 2016, 05:52
Just follow the pattern:
2^2 = 4 (1 to be multiple of 3) 2^3 = 8 (2 to be multiple of 3) 2^4 = 16 (1 to be multiple of 3) 2^5 = 32 (2 to be multiple of 3)
The pattern is subtracting 1 or 2. 2^256 is even, thus you must subtract 1 for a multiple of 3.



Manager
Joined: 03 Sep 2018
Posts: 57

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
28 Feb 2019, 01:42
stonecold, chetan2u, Abhishek009Thank you for the binomial method, indeed very helpful. I was just wondering what happens if the last term in the binomial is not even, e.g. if it would be xp1, would the remainder then be x1? Or can we only use the binomial method if the last term is positive (hence with even exponents)?
_________________
Please consider giving Kudos if my post contained a helpful reply or question.



Math Expert
Joined: 02 Aug 2009
Posts: 7425

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
28 Feb 2019, 03:14
ghnlrug wrote: stonecold, chetan2u, Abhishek009Thank you for the binomial method, indeed very helpful. I was just wondering what happens if the last term in the binomial is not even, e.g. if it would be xp1, would the remainder then be x1? Or can we only use the binomial method if the last term is positive (hence with even exponents)? The xp1 not understood. Please give some values ..
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html 4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentageincreasedecreasewhatshouldbethedenominator287528.html
GMAT Expert



Manager
Joined: 03 Sep 2018
Posts: 57

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
28 Feb 2019, 06:21
chetan2u wrote: ghnlrug wrote: stonecold, chetan2u, Abhishek009Thank you for the binomial method, indeed very helpful. I was just wondering what happens if the last term in the binomial is not even, e.g. if it would be xp1, would the remainder then be x1? Or can we only use the binomial method if the last term is positive (hence with even exponents)? The xp1 not understood. Please give some values .. Thanks for the quick reply, so assuming the value would have been \(4^{525}\) and we want to know the remainder when divided by 3, we could then say \((41)^{525}\), and the remainder would be (31)=2, is that correct? And can we do that in any such case?
_________________
Please consider giving Kudos if my post contained a helpful reply or question.



Math Expert
Joined: 02 Aug 2009
Posts: 7425

Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
Show Tags
28 Feb 2019, 09:28
ghnlrug wrote: chetan2u wrote: ghnlrug wrote: stonecold, chetan2u, Abhishek009Thank you for the binomial method, indeed very helpful. I was just wondering what happens if the last term in the binomial is not even, e.g. if it would be xp1, would the remainder then be x1? Or can we only use the binomial method if the last term is positive (hence with even exponents)? The xp1 not understood. Please give some values .. Thanks for the quick reply, so assuming the value would have been \(4^{525}\) and we want to know the remainder when divided by 3, we could then say \((41)^{525}\), and the remainder would be (31)=2, is that correct? And can we do that in any such case? If it is \(4^{525}\), we can write it as \((3+1)^{525}\)... so the remainder will be \(1^{525}\), which is 1.. But If it is \(2^{525}\), we can write it as \((31)^{525}\)... so the remainder will be \((1)^{525}\), which is 1. But the remainder cannot be negative, so remainder will be 31 or 2. This would be the case even when \(4^{525}\) is divided by 5, we can write it as \((51)^{525}\)... so the remainder will be \((1)^{525}\), which is 1. Thus the remainder here will be 51 or 4.
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html 4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentageincreasedecreasewhatshouldbethedenominator287528.html
GMAT Expert




Re: Which of the following must be subtracted from 2^526 so that the resul
[#permalink]
28 Feb 2019, 09:28






