If x and y are two odd positive integer which of the following is not an odd number 

When \( x \) and \( y \) are two odd positive integers, let's examine the arithmetic operations to determine which will not result in an odd number:

1. **Sum:** \( x + y \)  
   The sum of two odd numbers is always even. This is because the sum of two odd numbers can be expressed as:
   \[
   x + y = (2m + 1) + (2n + 1) = 2(m + n + 1)
   \]
   which is clearly divisible by 2, hence even.

2. **Difference:** \( x - y \)  
   The difference of two odd numbers is also even. Similar to the sum:
   \[
   x - y = (2m + 1) - (2n + 1) = 2(m - n)
   \]
   which is also divisible by 2, hence even.

3. **Product:** \( x \times y \)  
   The product of two odd numbers is always odd. This is because:
   \[
   x \times y = (2m + 1) \times (2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1
   \]
   which is of the form \( 2k + 1 \), where \( k \) is an integer, making the product odd.

4. **Division:** \( \frac{x}{y} \) or \( \frac{y}{x} \)  
   Division of two odd numbers can yield either a rational number or, in some cases, a whole number, but it does not specifically guarantee being odd or even. However, if the division results in an integer and \( x = y \), it equals 1, which is odd. If \( x \neq y \), the quotient is not necessarily an integer, and its odd/even nature doesn't directly apply.

Given the above analysis:

- **\( x + y \) and \( x - y \)** are **not odd** numbers.

So, **the sum and difference** of two odd positive integers are **not odd** numbers. 

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What happens when you add or subtract an odd and an even number?
2. How does the parity of numbers affect multiplication and division?
3. Can the square of an odd number be even?
4. What is the result when you multiply an odd number by an even number?
5. How do you determine if the result of a division is odd or even?

**Tip:** Remember, the sum or difference of two odd integers always results in an even integer!

Explore why the sum and difference of two odd positive integers are not odd numbers. This problem delves into arithmetic operations involving odd and even numbers, providing a detailed explanation backed by number theory concepts. Suitable for students in Grades 6-8.

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