If a is an even positive integer and b is an odd positive integer, then which of the following statement is true?

Options

A. a (b - 1) is even
B. a(b - 1) is odd
C. (a - 1) (b - 1) Is even
D. (a - 1) b is even

Answer

To solve the problem, we need to analyze the properties of even and odd integers. Let's break it down step by step.

### Given:
- \( a \) is an even positive integer.
- \( b \) is an odd positive integer.

### Definitions:
1. An **even integer** can be expressed as \( a = 2k \) where \( k \) is an integer.
2. An **odd integer** can be expressed as \( b = 2m + 1 \) where \( m \) is an integer.

### Options to Analyze:
1. \( ab - 1 \)
2. \( ab - 1 \) (seems to be a repetition, likely a typo)
3. \( (a - 1)(b - 1) \)
4. \( (a - 1)b \)

### Step-by-Step Analysis:

#### Step 1: Analyze \( ab - 1 \)
- Since \( a \) is even, \( a = 2k \).
- Since \( b \) is odd, \( b = 2m + 1 \).
- Therefore, \( ab = (2k)(2m + 1) = 2k(2m) + 2k = 2k(2m + 1) \), which is even.
- Thus, \( ab - 1 \) is even - 1, which is odd.

**Conclusion**: \( ab - 1 \) is **odd**.

#### Step 2: Analyze \( (a - 1)(b - 1) \)
- \( a - 1 \) is odd (since even - 1 = odd).
- \( b - 1 \) is even (since odd - 1 = even).
- The product of an odd number and an even number is even.

**Conclusion**: \( (a - 1)(b - 1) \) is **even**.

#### Step 3: Analyze \( (a - 1)b \)
- \( a - 1 \) is odd.
- \( b \) is odd.
- The product of two odd numbers is odd.

**Conclusion**: \( (a - 1)b \) is **odd**.

### Summary of Results:
1. \( ab - 1 \) is **odd**.
2. \( (a - 1)(b - 1) \) is **even**.
3. \( (a - 1)b \) is **odd**.

### Final Answer:
The true statements are:
- \( ab - 1 \) is odd.
- \( (a - 1)b \) is odd.