Can the Positive Integer p be Expressed as the Product of Two Integers GMAT Data Sufficiency

byRituparna Nath Content Writer at Study Abroad Exams

Question: Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?

(1) 31 < p < 37
(2) p is odd.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

“Can the positive integer p be expressed as the product of two integers, each of which is greater than 1?” – 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 2017". GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements.

This particular GMAT data sufficiency question assesses candidates’ critical thinking and hypervigilance. An abstract problem-solving question is mainly given and most of the difficulty comes from obtuse or clever wording, candidates usually miss it.

Solution and Explanation:

Approach Solution 1:

This question is very tricky question. Most candidates have been misinterpreting this.

Let us start by rephrasing the target question:
If an integer p can be expressed as the product of two integers, each of which is greater than 1, then that integer is a composite number (as opposed to a prime number). So . . . .
After rephrasing the target question, we get: Is integer p a composite number?

Aside: We have a video with tips on rephrasing the target question (below)

Statement 1: 31 < p < 37
There are 5 several values of p that meet this condition. Let's check them all.
p=32, which means p is a composite number
p=33, which means p is a composite number
p=34, which means p is a composite number
p=35, which means p is a composite number
p=36, which means p is a composite number
Since the answer to the rephrased target question is the SAME ("yes, p is a composite number") for every possible value of p, statement 1 is SUFFICIENT

Statement 2: p is odd
There are several possible values of p that meet this condition. Here are two:
Case a: p = 3 in which case p is not a composite number
Case b: p = 9 in which case p is a composite number
Since we cannot answer the rephrased target question with certainty, statement 2 is NOT SUFFICIENT

Correct Answer: A

Approach Solution 2:

Find Out: Can p be expressed as the product of two integers, each of which is greater than 1

Or is p = x*y, where x and y are greater than 1.

This means p can have the numbers that are not prime, since a prime number has only 2 factors: 1 and the number itself.
We will check each statement one by one:

Statement 1:
31 < p < 37
Values of p can be = 32, 33, 34, 35, 36
None of these is prime, hence p can be written as a product of x and y
Hence, the statement is sufficient.

Statement 2:
p is odd.
Odd numbers can both be prime and nonprime
Hence, it is not sufficient.

Correct Answer: A

Approach Solution 3:

As per the problem statement, we will evaluate the conditions and check if they satisfy.

Condition 1 - number that falls in given range are - 32 (2*16), 33 (3*11), 34 (2*17), 35 (7*5), 36 (2*18) .

Since, each number can be expressed as the product of two integers that are greater than 1 so answer is yes, it is sufficient.

Now, we will check the second one
Condition 2 - odd integer - 3 ->No
15 ->YES (3*5)

Since only condition 1 is sufficient to answer the given question and condition 2 does not answer the question so the answer is (A).

Correct Answer: A

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Can the Positive Integer p be Expressed as the Product of Two Integers GMAT Data Sufficiency

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