Q67. A Question is given followed by two Statements I and
II. Consider the Questions 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?

We need to find three distinct prime numbers (p1, p2, p3) whose sum (S) is also a prime number. First, consider the prime number 2. If 2 is one of the three distinct primes, say p1=2, then p2 and p3 must be odd primes.

The sum of two odd primes is always an even number. So, S = 2 + (odd + odd) = 2 + even = even. For S to be a prime number, it must be 2. However, the sum of three distinct positive primes cannot be 2.

Therefore, 2 cannot be one of the three distinct prime numbers. This means all three primes (p1, p2, p3) must be odd primes, and their sum (S) will also be an odd prime. **Statement I: Their sum is less than 23.** Since S must be an odd prime and S < 23, possible values for S are 3, 5, 7, 11, 13, 17, 19.

The smallest sum of three distinct odd primes is 3 + 5 + 7 = 15. So, S must be either 17 or 19. - If S = 17: We need three distinct odd primes that sum to 17. Possible combinations: (3, 5, 9 - 9 is not prime), (3, 7, 7 - not distinct).

No valid triplet sums to 17. - If S = 19: We need three distinct odd primes that sum to 19. The only combination is (3, 5, 11). (3+5+11=19, and 19 is prime). This is a unique triplet.

Thus, Statement I alone is sufficient to determine the three numbers as (3, 5, 11). **Statement II: One of the numbers is 5.** Let the three primes be 5, p2, p3. - If (3, 5, 11), sum = 19 (prime).

This is a valid triplet. - If (5, 7, 11), sum = 23 (prime). This is also a valid triplet. Since there are multiple possible triplets, Statement II alone is insufficient. Therefore, the question can be answered by using Statement I alone, but not Statement II alone.

This question tests data sufficiency and number theory, specifically properties of prime numbers.