To answer the following questions, which of the information given in the Statements (A), (B), (C) and (D) or (1), (2), (3) and (4) below is/are necessary/ sufficient?
P, Q, Rand S are four consecutive even integers. What is the value of the largest integer?
A. The average of the four numbers is the first prime number greater than 10.
B. The ratio between the largest and the smallest of the numbers is 7:4.
C. The squares of the sum of the smallest and the largest numbers is equal to the squares of the sum of the remaining two numbers.

Options

A. Any of them
B. Any two of them
C. Cand either A or B
D. Either A or B

Answer

To solve the problem, we need to find the value of the largest integer among four consecutive even integers, denoted as P, Q, R, and S. We will analyze the statements provided to determine if they are sufficient to answer the question.

### Step-by-Step Solution:

1. **Understanding Consecutive Even Integers**:
- Let the four consecutive even integers be represented as:
- P = x (the smallest even integer)
- Q = x + 2
- R = x + 4
- S = x + 6 (the largest even integer)

2. **Finding the Largest Integer**:
- The largest integer among P, Q, R, and S is S, which is equal to \( x + 6 \).

3. **Analyzing Statement A**:
- Statement A states: "The average of the four numbers is the first prime number greater than 10."
- The first prime number greater than 10 is 11.
- The average of the four integers is:
\[ \text{Average} = \frac{P + Q + R + S}{4} = \frac{x + (x + 2) + (x + 4) + (x + 6)}{4} = \frac{4x + 12}{4} = x + 3 \]
- Setting the average equal to 11 gives:
\[ x + 3 = 11 \implies x = 8 \]
- Therefore, the integers are 8, 10, 12, and 14. The largest integer S = 14.
- **Conclusion**: Statement A is sufficient.

4. **Analyzing Statement B**:
- Statement B states: "The ratio between the largest and the smallest of the numbers is 7:4."
- This means:
\[ \frac{S}{P} = \frac{7}{4} \]
- Substituting S and P gives:
\[ \frac{x + 6}{x} = \frac{7}{4} \]
- Cross-multiplying results in:
\[ 4(x + 6) = 7x \implies 4x + 24 = 7x \implies 3x = 24 \implies x = 8 \]
- Thus, the integers are again 8, 10, 12, and 14. The largest integer S = 14.
- **Conclusion**: Statement B is sufficient.

5. **Analyzing Statement C**:
- Statement C states: "The squares of the sum of the smallest and the largest numbers is equal to the squares of the sum of the remaining two numbers."
- This translates to:
\[ (P + S)^2 = (Q + R)^2 \]
- Substituting gives:
\[ (x + (x + 6))^2 = ((x + 2) + (x + 4))^2 \]
\[ (2x + 6)^2 = (2x + 6)^2 \]
- This is always true and does not help us find the specific values of P, Q, R, and S.
- **Conclusion**: Statement C is not sufficient on its own.

### Final Conclusion:
- Statements A and B are sufficient to determine the largest integer, while Statement C is not sufficient.