GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 22 Jan 2019, 01:06

### 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

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

## Events & Promotions

###### Events & Promotions in January
PrevNext
SuMoTuWeThFrSa
303112345
6789101112
13141516171819
20212223242526
272829303112
Open Detailed Calendar
• ### The winners of the GMAT game show

January 22, 2019

January 22, 2019

10:00 PM PST

11:00 PM PST

In case you didn’t notice, we recently held the 1st ever GMAT game show and it was awesome! See who won a full GMAT course, and register to the next one.
• ### Key Strategies to Master GMAT SC

January 26, 2019

January 26, 2019

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.

# If a and b are positive integers such that a < b, is b even?

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 27 Jun 2012
Posts: 371
Concentration: Strategy, Finance
Schools: Haas EWMBA '17
If a and b are positive integers such that a < b, is b even?  [#permalink]

### Show Tags

Updated on: 18 Dec 2012, 01:19
3
2
00:00

Difficulty:

35% (medium)

Question Stats:

71% (01:58) correct 29% (01:55) wrong based on 346 sessions

### HideShow timer Statistics

If a and b are positive integers such that a < b, is b even?

(1) $$\frac{b}{2}-\frac{a}{2}$$ is an integer.

(2) $$\frac{3b}{4}-\frac{a}{2}$$ is an integer.

_________________

Thanks,
Prashant Ponde

Tough 700+ Level RCs: Passage1 | Passage2 | Passage3 | Passage4 | Passage5 | Passage6 | Passage7
VOTE GMAT Practice Tests: Vote Here
PowerScore CR Bible - Official Guide 13 Questions Set Mapped: Click here
Finance your Student loan through SoFi and get $100 referral bonus : Click here Originally posted by PrashantPonde on 17 Dec 2012, 22:57. Last edited by Bunuel on 18 Dec 2012, 01:19, edited 1 time in total. Edited tags. ##### Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 52368 Re: If a and b are positive integers such that a < b, is b even? [#permalink] ### Show Tags 18 Dec 2012, 01:24 10 9 If a and b are positive integers such that a < b, is b even? (1) $$\frac{b}{2}-\frac{a}{2}$$ is an integer --> $$\frac{b}{2}-\frac{a}{2}=integer$$ --> $$b-a=2*integer=even$$. From $$b-a=even$$ it follows that either both a and b are even or both odd. Not sufficient. (2) $$\frac{3b}{4}-\frac{a}{2}$$ is an integer --> $$\frac{3b}{4}-\frac{a}{2}=integer$$ --> $$3b-2a=4*integer=even$$. Since $$2a=even$$, then we have that $$3b-even=even$$ --> $$3b=even$$ --> $$b=even$$. Sufficient. Answer: B. _________________ ##### Most Helpful Community Reply Manager Joined: 08 Apr 2012 Posts: 118 Re: If a and b are positive integers such that a < b, is b even? [#permalink] ### Show Tags 17 Dec 2012, 23:23 3 2 PraPon wrote: If a and b are positive integers such that a < b, is b even? (1) $$\frac{b}{2}-\frac{a}{2}$$ is an integer. (2) $$\frac{3b}{4}-\frac{a}{2}$$ is an integer. We know that a, b are positive integers and that a<b (1) --> b/2 - a/2 is integer. This means BOTH a, b are either Even or ODD. Ex: a=1, b=3 ==> (b-a)/2 = 1 (Integer) (b is odd) Also, when a=2, b=4 ==> (b-a)/2 = 1 (Integer) (b is even) Not sufficient (2) --> 3b/4 - a/2 is an integer. This can be written as (3b-2a)/4 is an integer Again both 3b and 2a need to be BOTH together even or odd for the expression to be an integer and the difference must be a multiple of 4. But, 2a is always even. This means 3b needs to be even as well. 3xEven = Even. So b is even. Answer is B. _________________ Shouvik http://www.Edvento.com admin@edvento.com ##### General Discussion VP Status: Been a long time guys... Joined: 03 Feb 2011 Posts: 1109 Location: United States (NY) Concentration: Finance, Marketing GPA: 3.75 Re: If a and b are positive integers such that a < b, is b even? [#permalink] ### Show Tags 17 Dec 2012, 23:33 PraPon wrote: If a and b are positive integers such that a < b, is b even? (1) $$\frac{b}{2}-\frac{a}{2}$$ is an integer. (2) $$\frac{3b}{4}-\frac{a}{2}$$ is an integer. Statement 1 leads us to two possibilities: i) either the two i.e.$$a/2$$ and $$b/2$$ are of the form x.5 AND y.5 respectively ii) both of them to be integers. Both of these possibilities lead us to the integer form. So B CAN be even and CANNOT be even. Statement 2- We have to keep in mind that a and b are integers. Hence $$3b/4$$ can only be an integer or of the form x.75 or x.25 Since (xx.75 or xx.25)-yy.5 CANNOT be an integer, therefore B has to be an integer such that $$3b/4$$ is an integer. For that to happen, B has to be a multiple of 4. The only possibility when 3b/4 falls in the form of xx.5 is when B is an even integer. OR $$a/2$$ can only be either entirely an integer or of the form yy.5. So in such cases, $$3b/4 - a/2$$ will be an integer ONLY when b is an even integer. Hence B. _________________ Senior Manager Joined: 27 Jun 2012 Posts: 371 Concentration: Strategy, Finance Schools: Haas EWMBA '17 Re: If a and b are positive integers such that a < b, is b even? [#permalink] ### Show Tags 18 Dec 2012, 08:39 Nothing can be simpler than Bunuel's explanation!! Thanks Bunuel!! I pretty much didn't like Manhattan GMAT's explanation. I guess it was too lengthy and convoluted. Manhattan GMAT explanation:- For this yes/no question, or goal is to try to find a definitive answer: either b is always even or b is always something other than even (odd or a fraction / decimal). If b is even only some of the time, then that information would be insufficient to answer the question. This is also a theory question; on such questions, we can try numbers or we can use theory. We can also test some numbers initially in order to help ourselves figure out or understand the theory more thoroughly and then use theory to help guide us through the rest of the problem. (1) INSUFFICIENT: We can test cases here to get started. First, let’s test the case where both a and b are even. If b = 4 and a = 2, then b/2-a/2=2-1=1. This makes sense using theory: we know that dividing an even integer by 2 will result in another integer. The variables a and b are both integers, so dividing each one by two will also yield integers, and one integer minus another integer will yield a third integer. Using both real numbers and theory, we have proved that the result will be an integer, so it’s possible for b to be even. Could b also be odd? Dividing an odd number by 2 yields some integer followed by the decimal 0.5 (for example 3/2 = 1.5). If we subtract one x.5 number from another, we’ll still get an integer. For instance, if b = 5 and a = 3, then b/2-a/2 = 2.5 – 1.5 = 1. It’s also possible, then, for b to be odd. Since b can be either even or odd, this statement is not sufficient. We have also now picked up something useful about the theory: an integer minus an integer will yield another integer. A non-integer minus another non-integer with the same decimal value (e.g., 2.5 – 1.5) will also yield an integer. (2) SUFFICIENT: We’re going to test even and odd cases here again. We already determined during statement 1 that a/2 will be an integer if a is even. What would need to be true in order for 3b/4 to be an integer as well? The value of b would have to be some multiple of 4 (in order to “cancel out” the 4 on the bottom of the fraction). We can try the same numbers we tried last time: b = 4 and a = 2. In this case, 3b/4-a/2 = 3 – 1 = 2. It’s possible, then, for b to be even. Can b be odd? There are two possible cases to test: odd b and odd a, or odd b and even a. An even value for a will result in an integer for a/2; for this to make statement 2 true, we would need 3b/4 to be an integer as well. 3b/4 will never result in an integer when b is odd, however, because an odd divided by an even will never be an integer. For example, if b = 5 and a = 2, then 3b/4-a/2 = 15/4 – 1 = not an integer. We can dismiss the case where a is even and b is odd. What about the case where both a and b are odd? If a is odd, then a/2 will be some number ending in 0.5. Can we make 3b/4 also end in 0.5, so that we’ll get an integer when subtracting the two? Let's try some odd positive integer possibilities for b: 3b/4 could equal 3/4, 9/4, 15/4, and so on, or the decimal equivalents 0.75, 2.25, 3.75, and so on. The pattern here alternates between 0.75 and 0.25; we cannot get 0.5. We can’t, then, get an integer value for 3b/4-a/2 as long as b is odd. The correct answer is B. _________________ Thanks, Prashant Ponde Tough 700+ Level RCs: Passage1 | Passage2 | Passage3 | Passage4 | Passage5 | Passage6 | Passage7 Reading Comprehension notes: Click here VOTE GMAT Practice Tests: Vote Here PowerScore CR Bible - Official Guide 13 Questions Set Mapped: Click here Finance your Student loan through SoFi and get$100 referral bonus : Click here

Board of Directors
Joined: 01 Sep 2010
Posts: 3279
Re: If a and b are positive integers such that a < b, is b even?  [#permalink]

### Show Tags

18 Dec 2012, 09:35
Bunuel wrote:
If a and b are positive integers such that a < b, is b even?

(1) $$\frac{b}{2}-\frac{a}{2}$$ is an integer --> $$\frac{b}{2}-\frac{a}{2}=integer$$ --> $$b-a=2*integer=even$$. From $$b-a=even$$ it follows that either both a and b are even or both odd. Not sufficient.

(2) $$\frac{3b}{4}-\frac{a}{2}$$ is an integer --> $$\frac{3b}{4}-\frac{a}{2}=integer$$ --> $$3b-2a=4*integer=even$$. Since $$2a=even$$, then we have that $$3b-even=even$$ --> $$3b=even$$ --> $$b=even$$. Sufficient.

Straight the same reasoning. Good
_________________
Intern
Joined: 17 Nov 2011
Posts: 1
If a and b are positive integers such that a < b, is b even?  [#permalink]

### Show Tags

22 Mar 2013, 14:27
If a and b are positive integers such that a < b, is b even?

(1) B/2- A/2 is an integer.

(2) 3*B/4 - A/2 is an integer
Math Expert
Joined: 02 Sep 2009
Posts: 52368
Re: If a and b are positive integers such that a < b, is b even?  [#permalink]

### Show Tags

22 Mar 2013, 14:30
freespiritfox wrote:
If a and b are positive integers such that a < b, is b even?

(1) B/2- A/2 is an integer.

(2) 3*B/4 - A/2 is an integer

Merging similar topics. Please refer to the solutions above.
_________________
VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1060
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Re: If a and b are positive integers such that a < b, is b even?  [#permalink]

### Show Tags

22 Mar 2013, 14:34
freespiritfox wrote:
If a and b are positive integers such that a < b, is b even?

(1) B/2- A/2 is an integer.

(2) 3*B/4 - A/2 is an integer

(1) B/2- A/2 is an integer.
$$B/2- A/2=i$$
$$B- A=2i$$ 2i is even and can be obtained as Even-Even or Odd-Odd so 1 is not sufficient

(2) 3*B/4 - A/2 is an integer
$$3*B/4 - A/2=i$$
$$3*B - 2A= 4i$$
4i is even and can be obtained as Even-Even or Odd-Odd.
Now consider that the second term is even (2A) so the other must be even also.
So, 3*B is even; can we say that B is also even?
The answer is yes, because Even = Odd*Even = 3(odd)*B(even)

So 2 is sufficient
_________________

It is beyond a doubt that all our knowledge that begins with experience.

Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6826
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If a and b are positive integers such that a < b, is b even?  [#permalink]

### Show Tags

10 Nov 2015, 10:03
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If a and b are positive integers such that a < b, is b even?

(1) b2 −a2 is an integer.

(2) 3b4 −a2 is an integer.

There are 2 variables (a,b) and 2 equations are given from the 2 conditions, so there is high chance (C) will be our answer.
Looking at the conditions,
Condition 1) b-a=2int=2int=even
Condition 2) 3b-2a=4int=even
As 2a=even, 3b-even=even and 3b=even, b=even, a=even. This answers the question 'yes' so this is sufficient and (C) seems to be the answer, but this is a commonly made mistake;
Looking at condition 1 again, b-a=even, so the question is answered 'yes' if b=4, a=2, but 'no' when b=3, a=1. So this is insufficient.
Looking at condition 2, 3b-2a=even, 3b=even-2a=even-even=even b=even. This answers the question 'yes' so this is sufficient. The answer is therefore (B).

For cases where we need 2 more equation, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only \$149 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

Current Student
Joined: 12 Aug 2015
Posts: 2626
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: If a and b are positive integers such that a < b, is b even?  [#permalink]

### Show Tags

01 Jan 2017, 09:22
Excellent Question.
Here is what i did in this question.

A and B are positive integers .
We need to see if B is even or not.

Statement 1
A=4
B=10
YES
A=7
B=13
NO

Hence Sufficient

Statement 2->
Here if B is odd => 3B/4 will be of the form x.25 or x.75 for integer x
And we cannot make A/2 of that form
Hence B cannot be odd
Hence B must be even

Hence Sufficient

Hence B

_________________

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 GMAT-Quant(700+)

AVERAGE GRE Scores At The Top Business Schools!

Intern
Joined: 07 May 2015
Posts: 37
Re: If a and b are positive integers such that a < b, is b even?  [#permalink]

### Show Tags

05 Aug 2018, 07:40
i could be overthinking this, but I was looking at manhattan gmat's explanation for B where they picked numbers, for my cases, what if b=6 and a=5?

3(6)/4 - (5)/2 = 4.5-2.5=1....? what is wrong here?
Re: If a and b are positive integers such that a < b, is b even? &nbs [#permalink] 05 Aug 2018, 07:40
Display posts from previous: Sort by