A Question is given followed by two Statements I and II. Consider the Question and the Statements. There are three distinct prime numbers whose sum is a prime number. Question : What are those three numbers? Statement-I : Their sum is less than 23. Statement-II : One of the numbers is 5. Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone ✓ UPSC's answer
(b) The Question can be answered by using either Statement alone
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
(d) The Question cannot be answered even by using both the Statements together

Why the answer is (a)

• Statement I states the sum of three distinct primes is less than 23. The smallest sum of three distinct primes is 2 + 3 + 5 = 10. Other combinations include 2+3+7=12, 2+3+11=16, 2+3+13=18, 2+3+17=22, 2+5+7=14, 2+5+11=18, 2+5+13=20, 2+7+11=20, 3+5+7=15, 3+5+11=19, 3+5+13=21, 3+7+11=21. Since multiple sets satisfy this condition, Statement I alone is insufficient.

• Statement II states one of the numbers is 5. The other two must be distinct primes different from 5. Possible pairs include (2,3) summing to 10, (2,7) summing to 14, (2,11) summing to 18, (2,13) summing to 20, (3,7) summing to 15, (3,11) summing to 19, (3,13) summing to 21, (7,11) summing to 23 (not prime), etc. Since multiple sets satisfy this condition, Statement II alone is insufficient.

• Combining both statements: The sum must be less than 23 and one number is 5. The possible sums involving 5 are 10 (2,3,5), 14 (2,5,7), 18 (2,5,11 or 3,5,10-invalid), 20 (2,5,13), 15 (3,5,7), 19 (3,5,11), 21 (3,5,13). All these sums (10, 14, 18, 20, 15, 19, 21) are prime numbers. Thus, there are still multiple valid sets of three distinct primes (e.g., {2,3,5}, {2,5,7}, {3,5,7}, etc.).

• Wait, let me re-evaluate the 'prime sum' constraint carefully. The question asks 'What are those three numbers?'. If multiple sets exist, the question cannot be answered uniquely.

• Let's re-read the options. Option (a) says 'can be answered by using one of the Statements alone'. This implies one statement yields a unique answer. Let's re-check Statement I. Sums < 23. Primes: 2,3,5,7,11,13,17,19. Distinct primes. Sum is prime.

• If Statement I is used alone: Sums < 23. Possible sets: {2,3,5} sum 10 (not prime). {2,3,7} sum 12 (not prime). {2,3,11} sum 16 (not prime). {2,3,13} sum 18 (not prime). {2,3,17} sum 22 (not prime). {2,5,7} sum 14 (not prime). {2,5,11} sum 18 (not prime). {2,5,13} sum 20 (not prime). {2,7,11} sum 20 (not prime). {3,5,7} sum 15 (not prime). {3,5,11} sum 19 (prime). {3,5,13} sum 21 (not prime). {3,7,11} sum 21 (not prime). {5,7,11} sum 23 (not < 23).

• Only {3,5,11} has a prime sum (19) and sum < 23. So Statement I alone gives a unique answer: 3, 5, 11.

• Let's re-check Statement II alone: One number is 5. Other two distinct primes. Sum is prime. Pairs for other two: (2,3) sum 10 (not prime). (2,7) sum 14 (not prime). (2,11) sum 18 (not prime). (2,13) sum 20 (not prime). (2,17) sum 24 (not prime). (3,7) sum 15 (not prime). (3,11) sum 19 (prime). (3,13) sum 21 (not prime). (7,11) sum 23 (prime). (7,13) sum 25 (not prime). (11,13) sum 29 (prime).

• So Statement II allows sets {3,5,11} (sum 19), {5,7,11} (sum 23), {5,11,13} (sum 29). Multiple answers. So Statement II alone is insufficient.

• Therefore, Statement I alone is sufficient, Statement II alone is not. This matches Option (a).

Why the other options are wrong

(b) The Question can be answered by using either Statement alone: Statement II alone allows multiple valid sets such as {3,5,11}, {5,7,11}, and {5,11,13}, so it cannot be answered using either statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be…: Statement I alone is sufficient to uniquely identify the set {3,5,11}, so the condition that it cannot be answered using either statement alone is false.
(d) The Question cannot be answered even by using both the Statements together: The question can be answered using Statement I alone, so it is incorrect to say it cannot be answered even with both statements.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.