Question

Is integer \(p\) a prime number? Statements: (I) \(2p\) has exactly 3 factors. (II) \(p\) is an even number.

• A. Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
• B. Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
• C. Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
• D. Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.

Hint:

A number with exactly 3 factors is always of the form \(p^2\), where \(p\) is prime.

Answer 1

The Correct Option is A

Concept: A number having exactly 3 factors must be the square of a prime number. For example: \[ 4=2^2 \] has factors: \[ 1,2,4 \] So exactly 3 factors. Prime numbers have exactly 2 factors.

Step 1: Checking Statement (I).
Given: \[ 2p \text{ has exactly 3 factors} \] So \(2p\) must be a square of a prime. Since \(2p\) is even, the only possible even square of a prime with exactly 3 factors is: \[ 4=2^2 \] Thus: \[ 2p=4 \] \[ p=2 \] Since \(2\) is prime, the answer is definitely YES. Thus, Statement (I) alone is sufficient.

Step 2: Checking Statement (II).
Given: \[ p \text{ is even} \] Possible even values: \[ 2,4,6,8,\dots \] Among these, only \(2\) is prime. Others are not prime. So we cannot determine uniquely. Thus, Statement (II) alone is not sufficient. Hence, Statement (I) alone is sufficient.