if a and b are two odd distinct prime numbers and if a > b then `a^2 - b^2` can never be divided by :

Options

A. a. 13
B. b. 11
C. c. 17
D. d. None of these

Answer

To solve the problem, we need to analyze the expression \( a^2 - b^2 \) where \( a \) and \( b \) are two distinct odd prime numbers and \( a > b \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression \( a^2 - b^2 \) can be factored using the difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]

**Hint**: Remember that the difference of squares can be factored into two binomials.

2. **Identifying the Properties of \( a \) and \( b \)**:
Since both \( a \) and \( b \) are odd prime numbers, we know that:
- Both \( a \) and \( b \) are odd.
- The difference \( a - b \) is even (since odd - odd = even).
- The sum \( a + b \) is also even (since odd + odd = even).

**Hint**: Odd numbers always produce even results when added or subtracted.

3. **Analyzing the Factors**:
Since both \( a - b \) and \( a + b \) are even, their product \( (a - b)(a + b) \) is also even. Therefore, \( a^2 - b^2 \) is divisible by 2.

**Hint**: Check the parity of the numbers involved to determine divisibility by 2.

4. **Checking Divisibility by Other Numbers**:
We need to check if \( a^2 - b^2 \) can be divisible by the given options. We can try specific odd prime numbers for \( a \) and \( b \) to see if \( a^2 - b^2 \) is divisible by certain primes.

- **Example 1**: Let \( a = 5 \) and \( b = 3 \):
\[ a - b = 5 - 3 = 2, \quad a + b = 5 + 3 = 8 \]
\[ a^2 - b^2 = 2 \times 8 = 16 \]
- Check divisibility by 2, 4, 8, etc.

- **Example 2**: Let \( a = 7 \) and \( b = 5 \):
\[ a - b = 7 - 5 = 2, \quad a + b = 7 + 5 = 12 \]
\[ a^2 - b^2 = 2 \times 12 = 24 \]
- Check divisibility by 3, 4, 6, etc.

**Hint**: Use specific values for \( a \) and \( b \) to test divisibility by the options provided.

5. **Conclusion**:
After testing various pairs of odd prime numbers, we find that \( a^2 - b^2 \) can be divisible by several primes, but we need to identify which one it cannot be divided by.

In this case, if we find that \( a^2 - b^2 \) is never divisible by a specific prime number (for example, 4), we conclude that is the answer.

**Final Answer**: The expression \( a^2 - b^2 \) can never be divided by **4**.