Let x and y be positive integers such that x is prime and y is composite. Then

Options

A. y-x cannot be an even integer
B. xy cannot be an even integer
C. `(x+y)//x` cannot be an even integer
D. None of these

Answer

To solve the problem, we need to analyze the conditions given: \( x \) is a prime number and \( y \) is a composite number. We will evaluate each option based on these definitions.

### Step-by-Step Solution:

1. **Understanding Prime and Composite Numbers**:
- A **prime number** is a natural number greater than 1 that has no positive divisors other than 1 and itself. The smallest prime number is 2.
- A **composite number** is a natural number greater than 1 that is not prime, meaning it has more than two distinct positive divisors. The smallest composite number is 4.

2. **Choosing Values for \( x \) and \( y \)**:
- Let’s take \( x = 2 \) (the smallest prime number).
- Let’s take \( y = 4 \) (the smallest composite number).

3. **Evaluating Each Option**:
- **Option A**: \( y - x \) cannot be an even integer.
- Calculation: \( y - x = 4 - 2 = 2 \).
- Since 2 is an even integer, this option is **false**.

- **Option B**: \( x \times y \) cannot be an even integer.
- Calculation: \( x \times y = 2 \times 4 = 8 \).
- Since 8 is an even integer, this option is **false**.

- **Option C**: \( \frac{x + y}{x} \) cannot be an even integer.
- Calculation: \( x + y = 2 + 4 = 6 \).
- Then, \( \frac{6}{2} = 3 \).
- Since 3 is an odd integer, this option is **true**.

- **Option D**: None of the above.
- Since we found that option C is true, option D is **false**.

4. **Conclusion**:
- The correct answer is **Option C**: \( \frac{x + y}{x} \) cannot be an even integer.