Consider the following statements in respect of two natural numbers p and q such that p is a prime number and q is a composite number : 1. p × q can be an odd number. 2. q/p can be a prime number. 3. p + q can be a prime number. Which of the above statements are correct?

(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3 ✓ UPSC's answer

Why the answer is (d)

• Statement 1 is correct: If p = 3 (prime) and q = 9 (composite), then p × q = 27, which is an odd number.

• Statement 2 is correct: If p = 2 (prime) and q = 6 (composite), then q/p = 3, which is a prime number.

• Statement 3 is correct: If p = 2 (prime) and q = 9 (composite), then p + q = 11, which is a prime number.

• Since all three statements can be true under specific valid conditions for natural numbers p and q, all statements are correct.

• Therefore, the option containing 1, 2, and 3 is the correct answer.

Why the other options are wrong

(a) 1 and 2 only: This option incorrectly excludes Statement 3, which is valid as shown by the example 2 + 9 = 11.
(b) 2 and 3 only: This option incorrectly excludes Statement 1, which is valid as shown by the example 3 × 9 = 27.
(c) 1 and 3 only: This option incorrectly excludes Statement 2, which is valid as shown by the example 6 / 2 = 3.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.