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

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.

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.

Straight the same reasoning. Good

(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

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.

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

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?

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

Remember the relation between the Variable Approach, and Common Mistake Types 3 and 4 (A and B)[Watch lessons on our website to master these approaches and tips]

Step 1: Apply Variable Approach(VA)

Step II: After applying VA, if C is the answer, check whether the question is key questions.

StepIII: If the question is not a key question, choose C as the probable answer, but if the question is a key question, apply CMT 3 and 4 (A or B).

Step IV: If CMT3 or 4 (A or B) is applied, choose either A, B, or D.

Let's apply CMT (2), which says there should be only one answer for the condition to be sufficient. Also, this is an integer question and, therefore, we will have to apply CMT 3 and 4 (A or B).

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find Is 'b' even? where 'a' and 'b' are positive integers such that a < b.

Second and the third step of Variable Approach: From the original condition, we have 2 variables (a and b). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 2 equations, C would most likely be the answer.

But we know that this is a key question [Integer question] and if we get an easy C as an answer, we will choose A or B.

Let’s take a look at each condition.

Condition(1) tells us that \(\frac{b}{2}\) - \(\frac{a}{2}\) is an integer.

=> \(\frac{b}{2}\) - \(\frac{a}{2}\) = integer

=> b - a = 2 * integer = that becomes EVEN.

=> ODD - ODD is also even and EVEN - EVEN is also even.

Since the answer is not a unique YES or NO, the condition is not sufficient by CMT 1.


Condition(2) tells us that \(\frac{3b}{4}\) - \(\frac{a}{2}\) = integer.

=> \(\frac{3b}{4}\) - \(\frac{a}{2}\) = integer

=> 3b - 2a = 4 * integer = that becomes EVEN.

=> 2a is always even and hence '3b' has to be even as EVEN - EVEN is even.

=> 3 * b = EVEN when 'b' is even - Is 'b' even - YES

Since the answer is a unique YES, the condition(2) alone is sufficient by CMT 1.


Condition (2) alone is sufficient.

So, B is the correct answer.

Answer: B

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