If p is a Prime Number Greater Than 2, What is the Value of p ? GMAT Data Sufficiency

Question: If p is a Prime Number Greater Than 2, What is the Value of p ?

(1) There are a total of 100 prime numbers between 1 and p+1
(2) There are a total of p prime numbers between 1 and 3,912

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: D

Solution and Explanation
Approach Solution 1:
The situation given is that p is a prime number greater than 2 and it is required to determine the value of p.

However, it is important to focus on the part that although it has been asked for the value of p, it is actually needed to determine whether the value of p is unique. Accordingly, from the given two conditions or statements in relevance to the question, it is to be determined that the value of p is unique. If the same is determined then the given statements would be considered sufficient. Any one of the statements or even both the statements can be sufficient or insufficient. Hence, it is important to judge and analyze both statements individually.

Statement One Alone:

There are a total of 100 prime numbers between 1 and p + 1.

If there are exactly 100 prime numbers between 1 and p + 1, then there are exactly 100 prime numbers in the list. This includes- 2, 3, 5, 7, 11, 13, …, p. Whatever value p is, p must be unique.

Accordingly, even if it is the 100th number in the list, p would be unique. Hence, Statement one alone can be considered sufficient in determining the value of p. Considering this aspect, the options, a, b and d can be eliminated.

Statement Two Alone:

The second statement states that- There are a total of p prime numbers between 1 and 3,912.

It is a fact that between two distinct positive integers, there must be a unique number of primes. For example, between 1 and 10 inclusive there are exactly 4 primes: 2, 3, 5, 7. There can’t be 3 primes or 5 primes between 1 and 10.

Therefore, the same logical explanation is applied to the second statement in order to find the value of p. Accordingly, it can be identified that between 1 and 3,912, there must be a number of prime numbers. These prime numbers must definitely be unique as well. Although the actual value of p cannot be determined from this statement, it can be considered that p must be unique. Hence, the second statement can also be considered to be absolutely sufficient as a separate and alone statement.

Hence, option C is the correct answer given that both the statements can be individually considered in answering the question.

Approach Solution 2:
(1) also sufficient: p is a prime number, so:
if p is the 100th prime, then there are 100 primes - viz., the first 100 primes - between 1 and p + 1.
if p is the 101th prime or later, then there are 101 or more primes, so that's no good.
if p is the 99th prime or earlier, then there are 99 or fewer primes; also no good.
therefore, p is the 100th prime.

(2) must be sufficient, as there is obviously some fixed number of primes between 1 and 3912. we don't care what that number is, because it's clear that there's only one such number (the number of primes in a fixed range isn't about to change anytime soon).

Approach Solution 3:
Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.
We have to find the value of 'p'
=> Given that 'p' is a prime number greater than '2'.

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

Let’s take a look at each condition.
Condition(1) tells us that there are a total of 100 prime numbers between 1 and p+1.
=> This means 'p' will be the 100^th prime number.
Since the answer is unique, the condition is sufficient by CMT 2.

Condition(2) tells us that there are a total of p prime numbers between 1 and 3,912
=> Listing all the prime numbers between 1 and 3,912 and then counting them will give us a value of 'p'
Since the answer is unique, the condition is sufficient by CMT 2.
Both conditions (1) and (2) alone are sufficient.
So, D is the correct answer.

“If p is a Prime Number Greater Than 2, What is the Value of p ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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If p is a Prime Number Greater Than 2, What is the Value of p ? GMAT Data Sufficiency

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