Two different prime number X and Y, both are greater than 2, then which of the following must be true ?

Options

A. A) `X-Y=23`
B. B) `X+Yne87`
C. C) Both (a) and (b)
D. D) None of these

Answer

To solve the problem, we need to analyze the properties of prime numbers greater than 2 and their sums and differences.

### Step-by-Step Solution:

1. **Identify the nature of prime numbers greater than 2**:
- All prime numbers greater than 2 are odd. This is because 2 is the only even prime number, and any other prime number must be odd (e.g., 3, 5, 7, 11, etc.).

**Hint**: Remember that prime numbers are defined as numbers greater than 1 that have no divisors other than 1 and themselves.

2. **Consider the difference of two odd prime numbers**:
- Let \( X \) and \( Y \) be two different prime numbers greater than 2. Since both are odd, their difference \( X - Y \) will also be even. This is because the difference between two odd numbers is always even.

**Hint**: Recall that odd - odd = even.

3. **Evaluate the statement \( X - Y = 23 \)**:
- Since 23 is an odd number, this statement cannot be true because we established that the difference of two odd numbers must be even.

**Hint**: Check if the result of the operation matches the expected parity (odd/even).

4. **Consider the sum of two odd prime numbers**:
- The sum \( X + Y \) of two odd numbers is always even. Therefore, if we are given that \( X + Y \) is not equal to an odd number (like 87), this statement must be true.

**Hint**: Remember that the sum of two odd numbers results in an even number.

5. **Conclusion**:
- Since the first statement \( X - Y = 23 \) is false (as it results in an odd number), and the second statement \( X + Y \neq 87 \) is true (as the sum of two odd numbers is even), we conclude that option B is correct.

### Final Answer:
The statement that must be true is: **\( X + Y \) is not equal to 87**.