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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

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.

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.

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.

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.

Answer: B.

Straight the same reasoning. Good
_________________

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.

Re: If a and b are positive integers such that a < b, is b even? [#permalink]

Show Tags

14 Sep 2014, 14:28

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.
_________________

Re: If a and b are positive integers such that a < b, is b even? [#permalink]

Show Tags

06 Nov 2015, 10:58

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.
_________________

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.
_________________

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