Step-by-step Solution:

Quick Solution

This problem tests the properties of odd and even numbers. The strategy is to determine the nature (odd or even) of the expressions in each statement based on the given conditions.

1. Establish the Basic Properties

From the problem statement, we know:

• x and y are ODD.
• z is EVEN.

Using these facts, we can determine the nature of the core components of the options:

• The expression (x - z) evaluates to (ODD - EVEN), which is always ODD.
• The expression (x - y) evaluates to (ODD - ODD), which is always EVEN.

2. Evaluate Each Statement

Now we can check each statement to see if it can ever be true.

• A. (x - z)²y is even
  • We know (x - z) is ODD, so (x - z)² = (ODD × ODD) = ODD.
  • We know y is ODD.
  • Therefore, the full expression (x - z)²y = (ODD × ODD) = ODD.
  • The result of this expression is always ODD. The statement that it is even is a contradiction and thus cannot be true.
• B. (x - z)² is odd
  • (x - z) is ODD. (ODD × ODD) = ODD. This statement is always true.
• C. (x - z)y is odd
  • (x - z) is ODD and y is ODD. (ODD × ODD) = ODD. This statement is always true.
• D. (x - y)² is even
  • (x - y) is EVEN. (EVEN × EVEN) = EVEN. This statement is always true.

Final Answer: The only statement that is impossible and can never be true is A. (x - z)²y is even.