GMAT Quant | Number Properties Q30

This GMAT quant practice questionis a data sufficiency question in Number Theory and Number Properties. Concept: properties of prime numbers and properties of multiples of 3. A GMAT 700 plus level question. GMAT hard math question.

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This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
3. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
4. EACH statement ALONE is sufficient to answer the question asked.
5. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Numbers

All numbers used are real numbers.

Figures

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

Note

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 30: Is the two digit positive integer P a prime number?

1. (P + 2) and (P - 2) are prime.
2. (P - 4) and (P + 4) are prime.

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Video Explanation

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Explanatory Answer

Step 1 of solving this GMAT DS question: Understand the Question Stem

What kind of an answer will the question fetch?
The question is an "Is" question. Answer to an "is" questions is either YES or NO.

When is the data sufficient?
The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

What additional information do we have about P from the question stem?
'P' is a 2-digit positive integer.

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Step 2 of solving this GMAT DS question:
Evaluate Statement (1) ALONE: (P + 2) and (P - 2) are prime.

Inference: (P - 2), P and (P + 2) are 3 consecutive odd integers.
Why?
Because (P - 2) and (P + 2) are prime, both numbers have to be odd.
(P - 2), P, and (P + 2) are three numbers in an arithmetic progression with a common difference of 2.
So, the 3 numbers have to be 3 consecutive odd or consecutive even integers. If (P - 2) and (P + 2) are odd, then these 3 numbers have to be 3 consecutive odd integers.

One out of 3 consecutive odd integers, (P - 2), P, and (P + 2) will definitely be a multiple of '3'.
If (P + 2) and (P - 2) are prime, then P has to be a multiple of '3', which is not prime.

The only exception is if the 3 consecutive odd numbers are 3, 5, and 7. However, we are dealing with two digit positive integers. So that possibility is ruled out.

Statement 1 ALONE is sufficient.
Eliminate choices B, C, and E. Choices narrow down to A or D.

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Step 3 of solving this GMAT DS question:
Evaluate Statement (2) ALONE: (P - 4) and (P + 4) are prime.

This is a brilliant statement.
1. The remainder when (P - 4) and (P - 1) are divided by 3 will be the same.
2. Similarly, the remainder when (P + 4) and (P + 1) are divided by 3 will be the same.
If (P - 4) and (P + 4) are prime, both (P - 4) and (P + 4) will leave a remainder when divided by 3.

Therefore, (P - 1) and (P + 1) will also leave a remainder when divided by 3. i.e., they are not divisible by 3.
(P - 1), P, (P + 1) are 3 consecutive positive integers.
One out of 3 consecutive integers, (P - 1), P, and (P + 1) will definitely be a multiple of '3'.
If (P - 1) and (P + 1) are not divisible by 3, then P has to be a multiple of '3'.
P cannot be 3 because P is a 2-digit number. So, that possiblity is ruled out.
Any 2-digit number that is a multiple of 3 cannot be prime.
Therefore, P is not prime.

Statement 2 ALONE is also sufficient.
Eliminate choice A.

Each statement is INDEPENDENTLY sufficient. Choice D is the answer.