8. If \( x \),is an even integer and \( y \) is an odd integer, which of the following must be an odd integer: a. \( 2 x+2 y \) b. \( 2 x-2 y \) (c) \( x+y+1 \) d. \( x+y+2 \) e. \( x-2 y \)

To determine which of the expressions must result in an odd integer given that \( x \) is an even integer and \( y \) is an odd integer, we start by analyzing each option. 1. \( 2x + 2y \): Since \( x \) is even, \( 2x \) is also even. \( 2y \) (where \( y \) is odd) is even because multiplying an odd integer by 2 results in an even integer. The sum of two even integers is even. So, this expression is even. 2. \( 2x - 2y \): Similar to the previous expression, \( 2x \) and \( 2y \) are both even. The difference between two even integers is even. Therefore, this expression is also even. 3. \( x + y + 1 \): Here, \( x \) (even) + \( y \) (odd) results in an odd integer (even + odd = odd). Adding 1 (which is odd) to an odd integer results in an even integer. So, this expression is even. 4. \( x + y + 2 \): Again, \( x \) (even) + \( y \) (odd) gives us an odd integer. Adding 2 (even) to an odd integer results in an odd integer. So, this expression is odd. 5. \( x - 2y \): \( x \) (even) minus \( 2y \) (even) results in an even integer since both are even. Hence, this expression is even. In conclusion, the only expression that must be an odd integer is: **Answer: (d) \( x + y + 2 \)**