It is currently 12 Dec 2017, 07:32

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

If a and b are positive integers such that a – b and a/b are

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

4 KUDOS received
Manager
Manager
avatar
Joined: 17 Aug 2009
Posts: 227

Kudos [?]: 301 [4], given: 25

If a and b are positive integers such that a – b and a/b are [#permalink]

Show Tags

New post 17 Dec 2009, 04:58
4
This post received
KUDOS
21
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

45% (01:44) correct 55% (01:35) wrong based on 664 sessions

HideShow timer Statistics

If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?

A. a/2
B. b/2
C. (a+b)/2
D. (a + 2)/2
E. (b+2)/2
[Reveal] Spoiler: OA

Kudos [?]: 301 [4], given: 25

Senior Manager
Senior Manager
avatar
Joined: 30 Aug 2009
Posts: 281

Kudos [?]: 194 [0], given: 5

Location: India
Concentration: General Management
Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 17 Dec 2009, 05:32
zaarathelab wrote:
If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?
A. a/2
B. b/2
C. (a+b)/2
D. (a + 2)/2
E. (b+2)/2


is the question correct?? none of the options seem to satisfy :|

here a>b since a/b is even integer

let a= 24 and b = 4.

A. a/2 = 12 => even
B. b/2 = 2 => even
C. (a+b)/2 = 14 => even
D. (a+2)/2 = 13 => odd
E. (b+2)/2 = 3 =>odd

now if a = 20 and b = 2 then a-b = 18 and a/b = 10
A. a/2 = 10 => even
B. b/2 = 1 => odd
C. (a+b)/2 = 11 => odd
D. (a+2)/2 = 13 => odd
E. (b+2)/2 = 2 => even

Kudos [?]: 194 [0], given: 5

Expert Post
20 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42564

Kudos [?]: 135345 [20], given: 12688

Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 17 Dec 2009, 05:38
20
This post received
KUDOS
Expert's post
11
This post was
BOOKMARKED
zaarathelab wrote:
If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?
A. a/2
B. b/2
C. (a+b)/2
D. (a + 2)/2
E. (b+2)/2


\(a-b\) even --> either both even or both odd

\(\frac{a}{b}\) even --> either both even or \(a\) is even and \(b\) is odd.

As both statements are true --> \(a\) and \(b\) must be even.

As \(\frac{a}{b}\) is an even integer --> \(a\) must be multiple of 4.

Options A is always even.
Options B can be even or odd.
Options C can be even or odd.
Options D: \(\frac{a+2}{2}=\frac{a}{2}+1\), as \(a\) is multiple of \(4\), \(\frac{a}{2}\) is even integer --> even+1=odd. Hence option D is always odd.
Options E can be even, odd.

Answer: D.
_________________

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?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 135345 [20], given: 12688

Senior Manager
Senior Manager
avatar
Joined: 30 Aug 2009
Posts: 281

Kudos [?]: 194 [0], given: 5

Location: India
Concentration: General Management
Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 17 Dec 2009, 05:41
1
This post was
BOOKMARKED
kp1811 wrote:
zaarathelab wrote:
If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?
A. a/2
B. b/2
C. (a+b)/2
D. (a + 2)/2
E. (b+2)/2


is the question correct?? none of the options seem to satisfy :|

here a>b since a/b is even integer

let a= 24 and b = 4.

A. a/2 = 12 => even
B. b/2 = 2 => even
C. (a+b)/2 = 14 => even
D. (a+2)/2 = 13 => odd
E. (b+2)/2 = 3 =>odd

now if a = 20 and b = 2 then a-b = 18 and a/b = 10
A. a/2 = 10 => even
B. b/2 = 1 => odd
C. (a+b)/2 = 11 => odd
D. (a+2)/2 = 13 => odd
E. (b+2)/2 = 2 => even


what the heck.... :x ...D it will be...sorry for oversight from my end :|

Kudos [?]: 194 [0], given: 5

Manager
Manager
avatar
Joined: 17 Aug 2009
Posts: 227

Kudos [?]: 301 [0], given: 25

Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 17 Dec 2009, 07:30
Quote:
As \frac{a}{b} is an even integer --> a must be multiple of 4.


Bunuel, you are an absolute whiz

Yep the catch is that a has to be a multiple of 4 for a/b to be even

OA is D

Kudos [?]: 301 [0], given: 25

Manager
Manager
avatar
Joined: 22 Jul 2009
Posts: 194

Kudos [?]: 554 [0], given: 6

Location: Manchester UK
Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 17 Dec 2009, 09:10
I didnt get why a has to be multiple of 4

Now if u take a as 4 and b as 2 then a-b is even a/b is even and a+b/2 is odd which is option C.

Please explain where i am going wrong?

Kudos [?]: 554 [0], given: 6

Senior Manager
Senior Manager
avatar
Joined: 30 Aug 2009
Posts: 281

Kudos [?]: 194 [0], given: 5

Location: India
Concentration: General Management
Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 17 Dec 2009, 09:20
sagarsabnis wrote:
I didnt get why a has to be multiple of 4

Now if u take a as 4 and b as 2 then a-b is even a/b is even and a+b/2 is odd which is option C.

Please explain where i am going wrong?


its a "MUST be" question. (a+b)/2 will not always be ODD.

also a-b and a/b is even and a and b will be even satisfying this condition. Since a/b is even so a will be multiple of 4.

Any set of values satisfying this will have a as 4n [n = natural number]

Kudos [?]: 194 [0], given: 5

Expert Post
3 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42564

Kudos [?]: 135345 [3], given: 12688

Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 17 Dec 2009, 09:35
3
This post received
KUDOS
Expert's post
1
This post was
BOOKMARKED
sagarsabnis wrote:
I didnt get why a has to be multiple of 4

Now if u take a as 4 and b as 2 then a-b is even a/b is even and a+b/2 is odd which is option C.

Please explain where i am going wrong?


First question: Why \(a\) has to be multiple of 4?

As we concluded \(a\) and \(b\) have to be even integers to meet both conditions in the stem: \(a=2m\) and \(b=2n\).

Then we have \(\frac{a}{b}\) is an even integer: \(\frac{a}{b}=2k\) --> \(a=2bk=4kn\) --> \(a\) is a multiple of \(4\).

Second question: Option C also COULD give an even result, so why not C?

The question asks: "which of the following MUST be an odd integer?"

If we take \(a=16\) and \(b=4\), then \(\frac{a+b}{2}=10=even\). So option C may or may not be odd and we are asked to determine which option is ALWAYS odd, hence C is out.

In fact:
Options A is always even.
Options B can be even or odd.
Options C can be even or odd.
Options D is always odd.
Options E can be even or odd.

Only option which is always odd is D.

Hope it helps.
_________________

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?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 135345 [3], given: 12688

Manager
Manager
avatar
Joined: 22 Jul 2009
Posts: 194

Kudos [?]: 554 [0], given: 6

Location: Manchester UK
Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 17 Dec 2009, 13:00
@Bunuel what a explanation man!!!!

Thanks a ton!!!!

Kudos [?]: 554 [0], given: 6

VP
VP
avatar
Joined: 17 Feb 2010
Posts: 1469

Kudos [?]: 804 [0], given: 6

Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 09 Aug 2010, 14:57
great explanation Bunuel.

Kudos [?]: 804 [0], given: 6

1 KUDOS received
Intern
Intern
User avatar
Joined: 20 Jul 2010
Posts: 49

Kudos [?]: 71 [1], given: 51

Re: If a and b are positive integers [#permalink]

Show Tags

New post 26 Oct 2010, 08:45
1
This post received
KUDOS
Given:
a > 0, b > 0 and a, b are Integers

a - b and a/b are both even intergers
==> \(a - b = 2m_1\)
\(\frac{a}{b} = 2m_2\) ==> \(a = (b)(2m_2)\)
==> a & b are both even, otherwise the above conditions won't be satisfied.

Now lets go with options one by one ...
A) a/2
==> can be even or odd ( a = 6 or a = 8)

B) b/2
==> Can be either even or odd ( b = 6 or b = 8)

C) (a+b) /2
==> Since, \(a = (b)(2m_2)\), \(\frac{a + b}{2}\) =\(\frac {(b)(2m_2) + b}{2}\)
==> \(\frac{a + b}{2}\) =\(\frac {(b)(2m_2 + 1)}{2}\)
==> Can be even or odd

D) (a+2)/2
==> \(\frac{a + b}{2}\) =\(\frac{2bm_2 + 2}{2}\)
==> \(\frac{a + b}{2}\) =\(bm_2 + 1\)
==> Since, b is even, \(bm_2\) is even and \(bm_2 + 1\) is odd always

E) (b+2)/2
==> \(\frac{b+2}{2} = 1 + \frac{b}{2}\)
==> Can be even or odd
_________________

Cheers!
Ravi


If you like my post, consider giving me some KUDOS !!!

Kudos [?]: 71 [1], given: 51

Manager
Manager
avatar
Joined: 25 Aug 2010
Posts: 90

Kudos [?]: 4 [0], given: 1

Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 27 Oct 2010, 22:40
good explanation from Bunuel

Kudos [?]: 4 [0], given: 1

Director
Director
User avatar
Status: No dream is too large, no dreamer is too small
Joined: 14 Jul 2010
Posts: 604

Kudos [?]: 1167 [0], given: 39

even [#permalink]

Show Tags

New post 21 Feb 2012, 23:11
a and b both have to be even to fulfill the condition.
a could be = 4, 6, 8 ...., b = 2,4, ....
so a - b = 2 (Even)
a/b= 2k (Even)
a/2=2k
a = 4k
so a must be multiple of 4
now try the options
ans. D
_________________

Collections:-
PSof OG solved by GC members: http://gmatclub.com/forum/collection-ps-with-solution-from-gmatclub-110005.html
DS of OG solved by GC members: http://gmatclub.com/forum/collection-ds-with-solution-from-gmatclub-110004.html
100 GMAT PREP Quantitative collection http://gmatclub.com/forum/gmat-prep-problem-collections-114358.html
Collections of work/rate problems with solutions http://gmatclub.com/forum/collections-of-work-rate-problem-with-solutions-118919.html
Mixture problems in a file with best solutions: http://gmatclub.com/forum/mixture-problems-with-best-and-easy-solutions-all-together-124644.html

Kudos [?]: 1167 [0], given: 39

Manager
Manager
avatar
Joined: 18 Oct 2010
Posts: 77

Kudos [?]: 195 [0], given: 26

Re: Even and Odd : GMATPrep [#permalink]

Show Tags

New post 21 May 2012, 06:13
kp1811 wrote:
zaarathelab wrote:
If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?
A. a/2
B. b/2
C. (a+b)/2
D. (a + 2)/2
E. (b+2)/2


is the question correct?? none of the options seem to satisfy :|

here a>b since a/b is even integer

let a= 24 and b = 4.

A. a/2 = 12 => even
B. b/2 = 2 => even
C. (a+b)/2 = 14 => even
D. (a+2)/2 = 13 => odd
E. (b+2)/2 = 3 =>odd

now if a = 20 and b = 2 then a-b = 18 and a/b = 10
A. a/2 = 10 => even
B. b/2 = 1 => odd
C. (a+b)/2 = 11 => odd
D. (a+2)/2 = 13 => odd
E. (b+2)/2 = 2 => even



small correction : (a+2)/2 = 11 as a= 20 ,so 22/2= 11 not 13

Kudos [?]: 195 [0], given: 26

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42564

Kudos [?]: 135345 [0], given: 12688

Re: If a and b are positive integers such that a – b and a/b are [#permalink]

Show Tags

New post 19 Jun 2013, 04:52

Kudos [?]: 135345 [0], given: 12688

1 KUDOS received
Verbal Forum Moderator
User avatar
Joined: 10 Oct 2012
Posts: 627

Kudos [?]: 1406 [1], given: 136

Premium Member
Re: If a and b are positive integers such that a – b and a/b are [#permalink]

Show Tags

New post 19 Jun 2013, 05:21
1
This post received
KUDOS
zaarathelab wrote:
If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?

A. a/2
B. b/2
C. (a+b)/2
D. (a + 2)/2
E. (b+2)/2


Given that (a-b) = even

Thus, \(b(\frac{a}{b}-1)\) = even --> as \(\frac{a}{b}\) is even, \((\frac{a}{b}-1)\) will be odd. Thus, b*odd = even--> b must be even

Thus, as a = b*2p --> a = 2l*2p = 4lp , where l,p are non-zero positive integers.

Thus,\(\frac{a}{2}\) + 1 = 2lp+1, which will always be an odd number.

D.
_________________

All that is equal and not-Deep Dive In-equality

Hit and Trial for Integral Solutions

Kudos [?]: 1406 [1], given: 136

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 14901

Kudos [?]: 287 [0], given: 0

Premium Member
Re: If a and b are positive integers such that a – b and a/b are [#permalink]

Show Tags

New post 21 Jul 2014, 05:44
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 287 [0], given: 0

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 14901

Kudos [?]: 287 [0], given: 0

Premium Member
Re: If a and b are positive integers such that a – b and a/b are [#permalink]

Show Tags

New post 26 Jul 2015, 22:56
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 287 [0], given: 0

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 14901

Kudos [?]: 287 [0], given: 0

Premium Member
Re: If a and b are positive integers such that a – b and a/b are [#permalink]

Show Tags

New post 20 Oct 2016, 08:14
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 287 [0], given: 0

Re: If a and b are positive integers such that a – b and a/b are   [#permalink] 20 Oct 2016, 08:14
Display posts from previous: Sort by

If a and b are positive integers such that a – b and a/b are

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.