Q. 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-1: 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.

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.

Correct answer: a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

##### Question from UPSC Prelims 2024 CSAT Paper

**Explanation:**

**1. Using Statement I alone:**

– We need to find three distinct prime numbers whose sum is prime and less than 23.

– The prime numbers less than 23 are 2, 3, 5, 7, 11, 13, 17, 19.

– After checking all possible combinations, we find that there is only one triplet that satisfies all conditions: (3, 5, 11)

– 3 + 5 + 11 = 19, which is prime and less than 23.

– This is the only solution that fits all criteria.

– Therefore, Statement I alone is sufficient to answer the question.

**2. Using Statement II alone:**

– We know one of the numbers is 5.

– We need to find two other distinct prime numbers that, when added to 5, result in a prime number.

– There are multiple possibilities, including:

a) 5 + 7 + 11 = 23 (prime)

b) 5 + 11 + 13 = 29 (prime)

– Since there’s more than one possible solution, Statement II alone is not sufficient to answer the question uniquely.

**Conclusion:**

The question can be answered using Statement I alone, as it leads to a unique solution (3, 5, 11).

The question cannot be answered using Statement II alone, as it leads to multiple possible solutions.

Therefore, the correct answer is: a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.