Properties of Odd Integers

CAT 1993 Slot 1 · QA · Easy · QA

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Given odd positive integers xx, yy and zz, which of the following is not necessarily true?

  1. A.

    x2y2z2x^2 y^2 z^2 is odd

  2. B.

    3(x2+y3)z23(x^2 + y^3)z^2 is even

  3. C.

    5x+y+z45x + y + z^4 is odd

  4. D.

    z2(x4+y4)2\frac{z^2 (x^4 + y^4)}{2} is even

Answer

D

Explanation

Let x=1,y=3,z=5x=1, y=3, z=5 (all odd): (a) 12×32×52=2251^2 \times 3^2 \times 5^2 = 225 (odd) (b) 3(1+27)×25=21003(1 + 27) \times 25 = 2100 (even) (c) 5(1)+3+625=6335(1) + 3 + 625 = 633 (odd) (d) 25(1+81)2=25×41=1025\frac{25(1 + 81)}{2} = 25 \times 41 = 1025 (odd, not even). Thus (d) is not necessarily true.

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