Understanding the Problem: Finding Distinct Prime Numbers

The question asks for a specific set of three distinct prime numbers whose sum is also a prime number. We are given two statements and need to determine if either statement alone, or both together, are sufficient to uniquely identify these three numbers.

Let the three distinct prime numbers be \(p_1, p_2, p_3\), where \(p_1 \neq p_2\), \(p_1 \neq p_3\), and \(p_2 \neq p_3\). Their sum is \(S = p_1 + p_2 + p_3\), and \(S\) must also be a prime number.

Let's consider the properties of prime numbers and their sums:

• Prime numbers are positive integers greater than 1 that have no positive divisors other than 1 and themselves. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, etc.
• The only even prime number is 2. All other prime numbers are odd.

Consider the sum \(S = p_1 + p_2 + p_3\):

• If one of the prime numbers is 2, let the set be \(\{2, p_2, p_3\}\), where \(p_2\) and \(p_3\) are distinct primes other than 2 (i.e., distinct odd primes). The smallest distinct odd primes are 3 and 5. So, \(p_2 \ge 3\) and \(p_3 \ge 5\) (assuming \(p_2 < p_3\)). The sum is \(S = 2 + p_2 + p_3\). Since \(p_2\) and \(p_3\) are odd, \(p_2 + p_3\) is an even number. Therefore, \(S = 2 + (\text{even number}) = \text{even number}\). For an even number to be prime, it must be 2. However, the smallest possible sum in this case is \(2 + 3 + 5 = 10\), which is greater than 2. Thus, the sum \(S\) cannot be a prime number if one of the primes is 2.
• If none of the prime numbers is 2, then all three primes \(p_1, p_2, p_3\) must be odd primes. The sum is \(S = \text{odd} + \text{odd} + \text{odd} = \text{even} + \text{odd} = \text{odd}\). An odd sum can be a prime number (e.g., 3, 5, 7, 11, 13, 17, 19, ...). The smallest sum of three distinct odd primes is \(3 + 5 + 7 = 15\). So, the sum \(S\) must be an odd prime number greater than or equal to 15.

From this analysis, we conclude that the three distinct prime numbers must all be odd primes, and their sum must be an odd prime \(S \ge 15\).

Analyzing Statement I: Sum is Less Than 23

Statement I says that the sum of the three distinct prime numbers is less than 23 (\(S < 23\)).

Based on our initial analysis, the sum \(S\) must be an odd prime number and \(S \ge 15\).

We need to find odd prime numbers \(S\) such that \(15 \le S < 23\). The odd primes in this range are 17 and 19.

Case 1: The sum \(S = 17\).

We need to find three distinct odd prime numbers \(p_1, p_2, p_3\) such that \(p_1 + p_2 + p_3 = 17\). Let's list distinct odd primes: 3, 5, 7, 11, 13, ... Let's assume \(p_1 < p_2 < p_3\).

• If \(p_1 = 3\), then \(p_2 + p_3 = 17 - 3 = 14\). We need two distinct odd primes \(p_2, p_3\) greater than 3 that sum to 14. Possible pairs of distinct odd numbers summing to 14 are (3, 11), (5, 9), (7, 7).
  • (3, 11): \(p_2 = 3\) is not greater than 3.
  • (5, 9): 9 is not prime.
  • (7, 7): Not distinct.
  No solution with \(p_1 = 3\).
• If \(p_1 = 5\), then \(p_2 + p_3 = 17 - 5 = 12\). We need two distinct odd primes \(p_2, p_3\) greater than 5 that sum to 12. The next odd prime after 5 is 7. If \(p_2 = 7\), then \(p_3 = 12 - 7 = 5\). This is not greater than 5 and not distinct from \(p_1\). No solution with \(p_1 = 5\).

No set of three distinct odd primes sums to 17.

Case 2: The sum \(S = 19\).

We need to find three distinct odd prime numbers \(p_1, p_2, p_3\) such that \(p_1 + p_2 + p_3 = 19\). Let's assume \(p_1 < p_2 < p_3\).

• If \(p_1 = 3\), then \(p_2 + p_3 = 19 - 3 = 16\). We need two distinct odd primes \(p_2, p_3\) greater than 3 that sum to 16. Possible pairs of distinct odd numbers summing to 16 are (3, 13), (5, 11), (7, 9).
  • (3, 13): \(p_2 = 3\) is not greater than 3.
  • (5, 11): Both 5 and 11 are distinct odd primes greater than 3 and 5 respectively. This gives the set \(\{3, 5, 11\}\). Sum \(3+5+11 = 19\), which is prime. This set works.
  • (7, 9): 9 is not prime.
  The set \(\{3, 5, 11\}\) is a solution with \(p_1 = 3\).
• If \(p_1 = 5\), then \(p_2 + p_3 = 19 - 5 = 14\). We need two distinct odd primes \(p_2, p_3\) greater than 5 that sum to 14. Possible pairs of distinct odd numbers summing to 14 are (3, 11), (5, 9), (7, 7).
  • (3, 11): \(p_2 = 3\) is not greater than 5.
  • (5, 9): \(p_2 = 5\) is not greater than 5, and 9 is not prime.
  • (7, 7): Not distinct.
  No solution with \(p_1 = 5\).

The only set of three distinct odd primes that sums to 19 is \(\{3, 5, 11\}\).

Since Statement I leads to exactly one possible set of three distinct prime numbers (\(\{3, 5, 11\}\)), Statement I alone is sufficient to answer the question.

Analyzing Statement II: One of the Numbers is 5

Statement II says that one of the three distinct prime numbers is 5.

Based on our initial analysis, the three distinct primes must be odd. So the set is of the form \(\{5, p_2, p_3\}\), where \(p_2\) and \(p_3\) are distinct odd prime numbers, neither of which is 5. The available odd primes other than 5 are 3, 7, 11, 13, 17, 19, 23, ...

The sum \(S = 5 + p_2 + p_3\) must be a prime number.

Let's find possible pairs of distinct odd primes \(\{p_2, p_3\}\) (neither being 5) and check if \(5 + p_2 + p_3\) is prime:

• Pair \(\{3, 7\}\): Sum \(5 + 3 + 7 = 15\) (not prime).
• Pair \(\{3, 11\}\): Sum \(5 + 3 + 11 = 19\) (prime). Set \(\{3, 5, 11\}\). This is a possible set.
• Pair \(\{3, 13\}\): Sum \(5 + 3 + 13 = 21\) (not prime).
• Pair \(\{3, 23\}\): Sum \(5 + 3 + 23 = 31\) (prime). Set \(\{3, 5, 23\}\). This is another possible set.
• Pair \(\{3, 29\}\): Sum \(5 + 3 + 29 = 37\) (prime). Set \(\{3, 5, 29\}\). This is yet another possible set.
• Pair \(\{7, 11\}\): Sum \(5 + 7 + 11 = 23\) (prime). Set \(\{5, 7, 11\}\). This is another possible set.
• Pair \(\{7, 17\}\): Sum \(5 + 7 + 17 = 29\) (prime). Set \(\{5, 7, 17\}\). This is another possible set.

Since Statement II allows for multiple possible sets of distinct prime numbers whose sum is prime (\(\{3, 5, 11\}\), \(\{3, 5, 23\}\), \(\{3, 5, 29\}\), \(\{5, 7, 11\}\), \(\{5, 7, 17\}\), etc.), Statement II alone is not sufficient to uniquely identify the three numbers.

Conclusion on Sufficiency

Statement I alone is sufficient because it uniquely identifies the three distinct prime numbers as 3, 5, and 11.

Statement II alone is not sufficient because it allows for multiple possible sets of three distinct prime numbers whose sum is prime.

Therefore, the question can be answered by using one of the statements alone (Statement I), but cannot be answered using the other statement alone (Statement II).

Revision Table: Key Concepts

Additional Information on Prime Number Properties

Understanding the behavior of prime numbers, especially concerning their sums, is crucial in number theory problems.

• The prime number 2 is unique because it is the only even prime. This makes it behave differently in sums compared to other primes.
• The sum of two odd numbers is always even.
• The sum of an odd and an even number is always odd.
• The sum of three odd numbers is always odd.
• The sum of one even and two odd numbers is always even.

In this problem, the sum of three distinct primes must be a prime number greater than 2 (since the smallest possible sum is \(2+3+5=10\)). A prime number greater than 2 must be odd. An odd sum of three numbers can only be obtained by adding three odd numbers. This immediately tells us that the three distinct prime numbers cannot include 2; they must all be odd.

This crucial deduction simplifies the problem greatly, as we only need to consider sets of distinct odd primes \(\{p_1, p_2, p_3\}\) such that their sum \(S = p_1 + p_2 + p_3\) is an odd prime number.

Statement I then helps narrow down the possible odd prime sums (\(< 23\)), and Statement II provides a specific constraint (one number is 5) to further test potential combinations.