Question

If the operation \( x \, ¤, y = 4x - y^2 \), and \( x, y \) are positive integers, which of the following cannot produce an odd value?

• A. \( x \, ¤, y^2 \)
• B. \( x \, ¤, 2y \)
• C. \( y \, ¤, x \)
• D. \( x \,¤, y \)
• E. \( x \, ¤, (y+1) \)

Hint:

When testing parity problems, check separately for even and odd substitutions.

Answer 1

The Correct Option is B

Step 1: Formula is \( 4x - y^2 \). 

Step 2: Check parities: 
- If \( y \) is odd, \( y^2 \) odd, \( 4x - y^2 \) = even - odd = odd. 
- If \( y \) is even, \( y^2 \) even, \( 4x - y^2 \) = even - even = even. 

Step 3: For \( x \,¤, 2y \), we substitute \( y' = 2y \). Then term = \( 4x - (2y)^2 = 4x - 4y^2 = 4(x-y^2) \), always even. 
Final Answer: \[ \boxed{x \,¤, 2y} \]