December 20, 2018 December 20, 2018 10:00 PM PST 11:00 PM PST This is the most inexpensive and attractive price in the market. Get the course now! December 22, 2018 December 22, 2018 07:00 AM PST 09:00 AM PST Attend this webinar to learn how to leverage Meaning and Logic to solve the most challenging Sentence Correction Questions.
Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 16 Feb 2010
Posts: 177

If 2a – b = 3c, where a, b, and c are nonzero integers
[#permalink]
Show Tags
Updated on: 09 Nov 2012, 01:25
Question Stats:
67% (02:30) correct 33% (02:59) wrong based on 428 sessions
HideShow timer Statistics
If 2a – b = 3c, where a, b, and c are nonzero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer? A. 2 B. 1 C. 1 D. 10 E. 12
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by zisis on 04 Sep 2010, 10:57.
Last edited by Bunuel on 09 Nov 2012, 01:25, edited 1 time in total.
Renamed the topic and edited the question.




Math Expert
Joined: 02 Sep 2009
Posts: 51302

Re: Puzzling question
[#permalink]
Show Tags
04 Sep 2010, 11:25
zisis wrote: If 2a – b = 3c, where a, b, and c are integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer?
Choices A 2 B 1 C 1 D 10 E 12
spent a fair amount of time on it and m stuck....please help \(2ab=3c\) > \(b=2a3c\). Now, \(average(a,b)=\frac{a+b}{2}=\frac{a+(2a3c)}{2}=3*\frac{ac}{2}=integer\), so average is a multiple of 3 (as given that average and all unknowns are integers). Only multiple of 3 among answers is 12. Answer: E. Hope it's clear.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Manager
Joined: 21 Aug 2010
Posts: 52
Location: United Kingdom
GMAT Date: 06282014

Re: Puzzling question
[#permalink]
Show Tags
04 Sep 2010, 13:40
Hi Bunuel,
I have a different approach to this question, Please let me know if It is correct...
Lets say Average of a & b is c
a+b/2=c a+b= 2c
2ab=3c a+b=2c  3a = 5c
a=5c/3
we can plug in all answer choices for average  " c " and see which one gives us an integer as "a" is also an integer.
5(2)/3 = 10/3 5(1)/3 = 5/3 5(1)/3 = 5/3 5(10)/3 = 50/3 5(12)/3 = 60/3= 20 ( Hence E 12 is the correct answer)



Manager
Joined: 19 Apr 2010
Posts: 176
Schools: ISB, HEC, Said

Re: Puzzling question
[#permalink]
Show Tags
06 Sep 2010, 04:47
I also got E by using substitution approach. However Bunuel suggested a good approach. Thanks



Manager
Joined: 15 Apr 2010
Posts: 116

Re: Puzzling question
[#permalink]
Show Tags
08 Sep 2010, 08:22
My solution: Code: 2a  b = 3c 2a + 2b  2b  b = 3c 2(a + b) = 3(b + c) (a + b)/2 = average = (3/4) (b + c)
For average to be an integer, it must be a multiple of 12. Hence E



Math Expert
Joined: 02 Sep 2009
Posts: 51302

Re: try this one
[#permalink]
Show Tags
06 Feb 2011, 14:05
Merging similar topics. bhandariavi wrote: If 2a – b = 3c, where a, b, and c are nonzero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer? A) 12 B) 3 c) 1 d) 1 E) 4 For the question above there are two correct answer A (for example a=9, b=15, c=1) and E (for example a=3, b=3, c=1), as both are multiple of 3.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



VP
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 1024

Re: Puzzling question
[#permalink]
Show Tags
19 May 2011, 00:04
2ab/3 = c integer.
a+b = 2 * average
average = 12 a+b = 24
3(a8)/3 = integer.
hence E



Manager
Joined: 12 Oct 2009
Posts: 150
Schools: Columbia, INSEAD, RSM, LBS

Re: Puzzling question
[#permalink]
Show Tags
19 May 2011, 20:11
E through substitution though its time consuming. Bunuel has the best approach



Manager
Joined: 27 Feb 2012
Posts: 119

Re: If 2a – b = 3c, where a, b, and c are nonzero integers, whi
[#permalink]
Show Tags
08 Nov 2012, 22:48
2013gmat wrote: If 2a – b = 3c, where a, b, and c are nonzero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer? A=2 B=1 C=1 D=10 E=12 checking options.... a+b = 24/20/2/2/4 b = a  24/20/2/2/4 Now, 2a  b = 3 c only 24 gives us a multiple of 3 that can be taken out. so E.
_________________

Please +1 KUDO if my post helps. Thank you.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8690
Location: Pune, India

Re: If 2a – b = 3c, where a, b, and c are nonzero integers
[#permalink]
Show Tags
04 Jun 2013, 20:12
zisis wrote: If 2a – b = 3c, where a, b, and c are nonzero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer?
A. 2 B. 1 C. 1 D. 10 E. 12 Responding to a pm: Bunuel has already given the algebraic approach which is quite simple and clear. I am guessing that since you are looking for another approach, you want me to solve it without using algebra. I instinctively jumped to the options in this question. They are the average of a and b so sum of a and b will be twice of the average so depending on which option we pick, the sum (a + b) will be 4 or 2 or 2 or 20 or 24. Note that a and b will be either both even or both odd since their sum must be even. The question says, which of the following could be the average i.e. there are probably many numbers that could be the average and one of them in included in this list. We also know that 2a – b = 3c Since right hand side has a 3, we know that 2a  b is divisible by 3. The easiest way to make it divisible by 3 is to make both a and b divisible by 3 which makes their sum divisible by 3 as well. Of the given options, only 24 is divisible by 3 hence (E) must be the answer.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



SVP
Joined: 06 Sep 2013
Posts: 1719
Concentration: Finance

Re: If 2a – b = 3c, where a, b, and c are nonzero integers
[#permalink]
Show Tags
22 Dec 2013, 17:59
zisis wrote: If 2a – b = 3c, where a, b, and c are nonzero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer?
A. 2 B. 1 C. 1 D. 10 E. 12 My way of solving 2a = 3c + b a = (3c + b)/2 So average (a + b)/2 We replace a and we get 3c+b/2, finally we get 6c + 3b = 3(2c+b) as the average So average must be a multiple of 3 and higher than 9 since numbers must be non zero integers Therefore only 12 fits the bill Hope it helps! Let me see that Kudos rain!!! Cheers J



SVP
Joined: 06 Sep 2013
Posts: 1719
Concentration: Finance

Re: If 2a – b = 3c, where a, b, and c are nonzero integers
[#permalink]
Show Tags
02 May 2014, 08:44
Here's actually another way of solving. First since all integers are >=0 it will be impossible to get a negative number so A,B are out right off the bat Now, we have that 2a  b = 3c and we need to find a+b / 2 So let's begin with answer choice E We have that a+b = 24 If we add 2a  b = 3c a + b = 24 We have 3a = 24 + 3c Now 24 + 3c is always a multiple of 3 so this one stays Let's try with D 2a  b = 3c a+b = 10 3a = 3c+10 Now 3c+10 won't ever be a multiple of 3 so OUT One final try, C 2a b = 3c a+b = 2 3a = 3c+2 Again, 3c + 2 will NEVER be a multiple of 3. Therefore only answer choice that is valid is E Gimme some freaking Kudos!!!! Cheers J



Director
Joined: 03 Feb 2013
Posts: 850
Location: India
Concentration: Operations, Strategy
GPA: 3.88
WE: Engineering (Computer Software)

Re: If 2a – b = 3c, where a, b, and c are nonzero integers
[#permalink]
Show Tags
18 Jul 2014, 23:19
Let N be the average so a+b = 2N 2a b = 3c Adding both the equation 3a = 2N + 3c As a,N,c are all integers, N has to be multiple of 3 to have the equation balanced. Hence 12  Option E)
_________________
Thanks, Kinjal My Debrief : http://gmatclub.com/forum/hardworknevergetsunrewardedforever189267.html#p1449379 My Application Experience : http://gmatclub.com/forum/hardworknevergetsunrewardedforever18926740.html#p1516961 Linkedin : https://www.linkedin.com/in/kinjaldas/
Please click on Kudos, if you think the post is helpful



Senior Manager
Joined: 28 Jun 2015
Posts: 292
Concentration: Finance
GPA: 3.5

Re: If 2a – b = 3c, where a, b, and c are nonzero integers
[#permalink]
Show Tags
19 Jul 2015, 06:32
\(2a  b = 3c\) \(2a  b  3c = 0\) \(3a + 3b + 3c = a + 4b + 6c\) \(\frac{(a+b+c)}{3} = a + 4b + 6c\) \((a+b+c) = 3(a + 4b + 6c)\) So, the average is a multiple of 3, hence Ans (E).
_________________
I used to think the brain was the most important organ. Then I thought, look what’s telling me that.



CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2711
Location: India
GMAT: INSIGHT
WE: Education (Education)

Re: If 2a – b = 3c, where a, b, and c are nonzero integers
[#permalink]
Show Tags
19 Jul 2015, 06:51
zisis wrote: If 2a – b = 3c, where a, b, and c are nonzero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer?
A. 2 B. 1 C. 1 D. 10 E. 12 We can always solve such questions by taking certain values for a, b and c keeping in mind that if average of a and b must be integers then both a and b must be either even or both must be odd Also 2a  b must be a multiple of 3Trying with Even Numbers first Let, a=8, b=4 i.e. 2ab = 164 = 12 = 3c i.e. c=4 i.e. Average of a and b = (8+4)/2 = 6 Obtained Average Relates with 12 so doubling every number in previous step i.e. Let, a=16, b=8 i.e. 2ab = 328 = 24 = 3c i.e. c=8 i.e. Average of a and b = (16+8)/2 = 12 SUCCESSAnswer: Option E
_________________
Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha email: info@GMATinsight.com I Call us : +919999687183 / 9891333772 Online OneonOne Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html
ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION



Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2830

Re: If 2a – b = 3c, where a, b, and c are nonzero integers
[#permalink]
Show Tags
26 Feb 2018, 10:19
zisis wrote: If 2a – b = 3c, where a, b, and c are nonzero integers, which of the following could be the average (arithmetic mean) of a and b, if the average must itself be an integer?
A. 2 B. 1 C. 1 D. 10 E. 12 We can manipulate the first equation to read: 2a  3c = b Now let’s set up an expression for the average of a and b: (a + b)/2 = ? (a + 2a  3c)/2 = ? (3a  3c)/2 = ? 3(a  c)/2 = ? Since 3 is not divisible by 2, so a  c must be divisible by 2. Therefore, since (a  c)/2 must be an integer and 3(a  c)/2 must be a multiple of 3. The only multiple of 3 in the answer choices is E, 12; thus, E is the correct answer. Answer: E
_________________
Jeffery Miller
Head of GMAT Instruction
GMAT Quant SelfStudy Course
500+ lessons 3000+ practice problems 800+ HD solutions




Re: If 2a – b = 3c, where a, b, and c are nonzero integers &nbs
[#permalink]
26 Feb 2018, 10:19






