If `a+b` is even, what is ab?

Answer

To solve the problem "If \( a + b \) is even, what is \( ab \)?", we will analyze the conditions under which the sum \( a + b \) is even and how it affects the product \( ab \).

### Step-by-Step Solution:

1. **Understanding Even and Odd Numbers**:
- An even number is defined as any integer that can be expressed in the form \( 2k \), where \( k \) is an integer (e.g., 0, 2, 4, 6, ...).
- An odd number is defined as any integer that can be expressed in the form \( 2k + 1 \) (e.g., 1, 3, 5, 7, ...).

2. **Condition Given**:
- We are given that \( a + b \) is even. This can happen in two scenarios:
- Both \( a \) and \( b \) are even.
- Both \( a \) and \( b \) are odd.

3. **Case 1: Both \( a \) and \( b \) are Even**:
- Let \( a = 2m \) and \( b = 2n \) for some integers \( m \) and \( n \).
- Then, the product \( ab = (2m)(2n) = 4mn \), which is even.

4. **Case 2: Both \( a \) and \( b \) are Odd**:
- Let \( a = 2m + 1 \) and \( b = 2n + 1 \) for some integers \( m \) and \( n \).
- Then, the product \( ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 \).
- This can be simplified to \( ab = 2(2mn + m + n) + 1 \), which is odd.

5. **Conclusion**:
- From the two cases, we conclude:
- If \( a + b \) is even and both \( a \) and \( b \) are even, then \( ab \) is even.
- If \( a + b \) is even and both \( a \) and \( b \) are odd, then \( ab \) is odd.
- Therefore, we cannot definitively determine whether \( ab \) is even or odd based solely on the information that \( a + b \) is even.

### Final Answer:
We cannot determine whether \( ab \) is even or odd based on the information that \( a + b \) is even.