If a and bare odd numbers, then which of the following is even ?

Options

A. a + b + ab
B. a+b-1
C. a + b +1
D. a + b + 2ab

Answer

To determine which of the given expressions is even when \( a \) and \( b \) are both odd numbers, we can analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Odd Numbers:**
- An odd number can be expressed in the form \( 2n + 1 \), where \( n \) is an integer.
- For example, if \( a \) and \( b \) are odd, we can write:
\[ a = 2m + 1 \quad \text{and} \quad b = 2n + 1 \]
for some integers \( m \) and \( n \).

2. **Calculating \( a + b \):**
- The sum of two odd numbers:
\[ a + b = (2m + 1) + (2n + 1) = 2m + 2n + 2 = 2(m + n + 1) \]
- This result is even since it is divisible by 2.

3. **Calculating \( ab \):**
- The product of two odd numbers:
\[ ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 \]
- This result is odd since it is of the form \( 2k + 1 \).

4. **Evaluating Each Option:**
- **Option 1: \( a + b + ab \)**
- We know \( a + b \) is even and \( ab \) is odd.
- Even + Odd = Odd.
- **Option 2: \( a + b - 1 \)**
- \( a + b \) is even, and subtracting 1 from an even number gives us an odd number.
- **Option 3: \( a + b + 1 \)**
- \( a + b \) is even, and adding 1 to an even number gives us an odd number.
- **Option 4: \( a + b + 2ab \)**
- We know \( a + b \) is even and \( ab \) is odd.
- \( 2ab \) is even (since it is \( 2 \times \text{(odd)} \)).
- Even + Even = Even.

5. **Conclusion:**
- The only expression that results in an even number is:
\[ a + b + 2ab \]
- Therefore, the correct answer is **Option 4**.