Two numbers a and b are such that one is odd and the other is even. Which statement is necessarily true?

Options

A. `a^2+b` is even
B. (a+b) is even
C. `a^bxxb^a` is even
D. `a^2-b^2` is even

Answer

To solve the question regarding two numbers \( a \) and \( b \) where one is odd and the other is even, we need to analyze the statements provided in the options. Let's go through the reasoning step by step.

### Step-by-Step Solution:

1. **Understanding Odd and Even Numbers**:
- An even number can be expressed as \( 2n \) where \( n \) is an integer (e.g., 0, 1, 2, ...).
- An odd number can be expressed as \( 2m + 1 \) where \( m \) is also an integer (e.g., 0, 1, 2, ...).

2. **Assigning Values**:
- Let's assign \( a = 2 \) (even) and \( b = 3 \) (odd) for our analysis.

3. **Evaluating Each Statement**:
- **Option 1**: \( a + b \) is odd.
- Calculation: \( 2 + 3 = 5 \) (which is odd).
- This statement is true, but we need to check if it is necessarily true for all cases.
- **Option 2**: \( a - b \) is even.
- Calculation: \( 2 - 3 = -1 \) (which is odd).
- This statement is false.
- **Option 3**: \( a \times b \) is even.
- Calculation: \( 2 \times 3 = 6 \) (which is even).
- This statement is true and will always hold since multiplying any even number by any other number (even or odd) results in an even number.
- **Option 4**: \( a - b \) is odd.
- Calculation: \( 2 - 3 = -1 \) (which is odd).
- This statement is true, but similar to option 1, we need to check if it is necessarily true for all cases.

4. **Conclusion**:
- The only statement that is necessarily true for any odd and even pair \( a \) and \( b \) is **Option 3**: \( a \times b \) is even.

### Final Answer:
The statement that is necessarily true is: **Option 3: \( a \times b \) is even.**