GMAT 700 Level Arithmetic Question 36

The given question is a GMAT Hard Math data sufficiency question. This question is a GMAT arithmetic practice question that tests your understanding of properties of positive integers from the topic Number Properties.

Question 36: What is the value of (m + n) if m and n are positive integers?

Statement 1: m² – n² = 105
Statement 2: Neither m nor n is divisible by 8.

Video Explanation

Explanatory Answer | GMAT Arithmetic DS Q36

Step 1: Evaluate Statement 1 ALONE

Statement 1: m² – n² = 105
m and n are positive integers.
m² – n² can be written as (m – n)(m + n). Therefore, (m – n)(m + n) = 105
Both (m – n) and (m + n) have to be positive integers. (Why? Because (m + n) is positive and the product of (m + n) and (m - n) is positive.)
The ways of expressing 105 as a product of 2 positive integers is listed in the table given below.

So, (m + n) could be 105, 35, 21, or 15.
We are not able to answer the question with a UNIQUE value.
Hence, statement 1 alone is not sufficient.
Eliminate answer options A and D.

Step 2: Evaluate Statement 2 ALONE

Statement 2: Neither m nor n is divisible by 8.
Infinite possibilities exist for this condition.

We are not able to answer the question with a UNIQUE value.
Hence, statement 2 alone is not sufficient.
Eliminate answer option B.

Step 3: Evaluate Statements TOGETHER

Statement 1: m² – n² = 105
Statement 2: Neither m nor n is divisible by 8.

From statement 1: (m – n) (m + n) = 105 and both (m – n) and (m + n) are positive integers.

From Statement 2: We can eliminate (19, 16) and (13, 8).
We still have two values for (m, n): (53, 52) and (11, 4)
Therefore, (m + n) could be 105 or 15.

Despite combining both the statements, we are not able to find a UNIQUE value for (m + n).
Statements TOGETHER are NOT sufficient.
Eliminate answer option C.

Choice E is the correct answer.