Parity of Odd Integer Operations

CAT 2000 Slot 1 · QA · Easy · Number Systems

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Let x,yx, y and zz be distinct integers, that are odd and positive. Which one of the following statements cannot be true?

  1. A.

    xyz2xyz^2 is odd

  2. B.

    (x−y)2z(x - y)^2 z is even

  3. C.

    (x+y−z)2(x+y)(x + y - z)^2 (x + y) is even

  4. D.

    (x−y)(y+z)(x+y−z)(x - y)(y + z)(x + y - z) is odd

Answer

D

Explanation

Since xx and yy are odd, (x−y)(x - y) is even. Product of an even number with any integer is always even. Therefore, (x−y)(y+z)(x+y−z)(x - y)(y + z)(x + y - z) must be even, so it can NEVER be odd.

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