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.

Correct Option

The question asks to identify three distinct prime numbers whose sum is also a prime number. Let these numbers be p₁, p₂, p₃, and their sum S = p₁ + p₂ + p₃. S must also be a prime number.

First, consider the nature of the prime numbers. If one of the primes is 2, then the sum S would be 2 + (odd prime) + (odd prime) = 2 + (even number) = (even number). Since S must be a prime number, and the only even prime number is 2, S would have to be 2. However, the sum of three distinct primes (even if one is 2) would be at least 2 + 3 + 5 = 10, which is greater than 2. Therefore, none of the three distinct prime numbers can be 2. This implies that all three prime numbers must be odd primes (e.g., 3, 5, 7, 11, 13, 17, 19, ...).

Evaluating Statement I: Their sum is less than 23.

• We are looking for three distinct odd primes p₁, p₂, p₃ such that p₁ + p₂ + p₃ = S, where S is prime and S < 23.
• Consider the smallest distinct odd primes: 3, 5, 7. Their sum is 15 (not prime).
• Next combination: 3, 5, 11. Their sum is 19. 19 is a prime number, and 19 < 23. This triplet (3, 5, 11) is a valid solution.
• Next combination: 3, 5, 13. Their sum is 21 (not prime).
• Next combination: 3, 7, 11. Their sum is 21 (not prime).
• Next combination: 3, 7, 13. Their sum is 23. 23 is a prime number, but the sum is not strictly less than 23.
• Any other combination of three distinct odd primes will result in a sum of 23 or greater. For example, (5, 7, 11) has a sum of 23.
• Thus, the only triplet satisfying the conditions and Statement I is (3, 5, 11).

Therefore, Statement I alone is sufficient to uniquely identify the three numbers.

Incorrect Options

Evaluating Statement II: One of the numbers is 5.

• We are looking for three distinct odd primes p₁, p₂, p₃, where one of them is 5, such that p₁ + p₂ + p₃ = S, where S is prime.
• Possible triplets satisfying this condition include:
  • (3, 5, 11): Sum = 19 (prime)
  • (5, 7, 11): Sum = 23 (prime)
  • (5, 7, 17): Sum = 29 (prime)
  • (5, 11, 13): Sum = 29 (prime)
• Since multiple triplets satisfy Statement II, it does not uniquely identify the three numbers.

Therefore, Statement II alone is not sufficient to answer the question.

Based on the analysis, Statement I alone is sufficient, but Statement II alone is not sufficient. This aligns with option 1.