If a and b are Positive Integers Such That a – b and a/b are GMAT Problem Solving

Question: 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/2
b/2
(a+b)/2
(a + 2)/2
(b+2)/2

Correct Answer: D

Solution and Explanation
Approach Solution 1:
The given case is that a and b are both positive integers in such a way that a-b and a/b both are even integers. This implies that a > 0 and b > 0

This can further imply that a-b is equal to \(2m_1\)
This can be equated as-

\(\Rightarrow a-b-2m_1\)

Further, the given case of a/b is also even integer this can be implied with a.b is equal to \(2m_2\)
This can be equated in terms of-n

\(\frac{a}{b}=2m_2\Rightarrow a =(b)(2m_2)\)

This significantly implicates that a and b are both even integers because if they were not, the above conditions would not have been satisfied.
Based on the given five options, to find the odd integer among them all, each of the options need to be evaluated. This can be done as follows:

Based on the above conditions given, a/2 can be even or can be odd. This implies a can be equal to 6 or a can be equal to 8. Hence, the results for both the integers when divided by 2 can either turn out to be an odd integer 3 or even integer 4.

Considering the given conditions above, similar to a, b can be an odd integer as well as an even integer. This implies that if b is equal to 6 then, the result would be an odd integer 3 and if b is 8 then the result with an even integer 4.

(a+b)/2

Based on this given option, the integer can be even or odd. This can be identified in terms of the fact where above it has been stated that a is equal to ((b)(\(2m_2\)). accordingly, if the option of (a+b)/2 has to be evaluated it can be done in the following equation base:

\(\Rightarrow\frac{a+b}{2} =\frac{(b)(2m_2)+b}{2}\)

\(\Rightarrow \frac{a+b}{2}= \frac{(b)(2m_2+1)}{2}\)

Hence, based on the above evaluation of the equation, the results can be considered to be identified if it can be even or odd.

(a+2)/2

Based on the given above situation as well as the evaluated values for a and b, (a+2)/2 needs to be understood to find the integer, if it is odd or even. This can be done through to sequence of equations. Accordingly, the following evaluation states that-

\(\Rightarrow\frac{a+b}{2}=\frac{2bm_2+2}{2}\)

\(\Rightarrow\frac{a+b}{2}=bm_2+1\)

Accordingly, it has been found that b is the even integer from the aspect of \(bm_2\) which is always even. However, when \(bm_2\) is added to 1, the results always turn out to be odd. Hence, \(bm_2+1\) is always odd.

(b+2)/2

Based on the above evaluation and given case scenario, it can be identified that the equation (b+2)/2 needs to be solved to understand whether the results are odd or even. Accordingly, the answer can be found with the following equation:

\(\Rightarrow\frac{b+2}{2}=1+\frac{b}{2}\)

Based on the equation that has been considered above, it can be evaluated that the results of the equation can be either even or odd. Hence, this equation might have both results.
From evaluating equations from all the five options to identify which option could yield an odd integer as asked in the question, option D, which is (a+2)/2 will always yield odd integers hence, the right answer.

Approach Solution 2:
Considering the given question scenario, it can be identified that evaluating all the given options would help in finding the odd integer based on the given five cases. These can be evaluated based on two important cases-

Case 1: a is even and b is even based on the given question. Accordingly, even/even will result with an even integer.
Case 2: a is even and b is odd. Accordingly, even/odd will result with an even integer.
When both options are combined, both a and b have to be even to satisfy both statements.
Hence, it can be stated that a has to be a multiple of 4 as it is the product of two even numbers and for b, it has to be an even integer or multiple of 2.

Now, evaluate the five options as follows:

Since a is a multiple of 4, a/2 is always even which hence, is not the right answer.

B is an even number although b can turn out to be odd depending on the value of b like 2/2 is equal to 1 or even with 4/2 is equal to 2.

(a+b)/2

This implies that (a+b ) will be an even integer and Even/2 can be odd or even as explained in option B. hence, option C cannot be the right answer.

(a + 2)/2

The above equation implies that, a/2 + 1. Since a is a multiple of 4, a/2 is always even. However, Even + 1 will always give an odd integer which means that the answer to the equation of D is always odd. Hence, D is the right answer.

(b+2)/2

The above equation states that b + 2 should be an Even number. Any number which is Even/2 can be odd or even hence, it cannot be the right answer.
Overall, D is the correct answer, because only (a+2)/2 can give only an odd integer.

Approach Solution 3:
1. a – b is even means that either both a and b are even OR both a and b are odd;
2. a/b is even means that either both a and b are even OR a is even and b is odd.

Since both statements are true, then both a and b must be even.
Next, since a/b is even AND both a and b are even, then a must be multiple of 4.

So, we have that: both a and b are even and a is a multiple of 4.

The question asks: which of the following must be an odd integer

A. a/2. Since a is a multiple of 4, then a/2 is always even.

B. b/2. Since b is even, then b/2 can be even or odd. For example, consider b = 4 and b = 2.

C. (a + b)/2. Since both a and b are even, then (a + b)/2 can be even or odd. For example, consider (a, b) = (8, 4) and (a, b) = (4, 2).

D. (a + 2)/2. Since a is a multiple of 4, then (a + 2)/2 = (4k + 2)/2 = 2k + 1 = even + 1 = odd. Hence option D is always odd.

E. (b + 2)/2. Since b is even, then (b + 2)/2 can be even or odd. For example, consider b = 2 and b = 4.

“If a and b are positive integers such that a – b and a/b are”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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