If x, y, and z are integers such that x + y + z = 0 and x^3 y - z is an odd integer, which… — GMAT

If $x$ , $y$ , and $z$ are integers such that $x + y + z = 0$ and $x^3 y - z$ is an odd integer, which of the following expressions MUST be an even integer?

$x y z$ Answer
B
$x + y$
C
$y + z$
D
$x - y$
E
$x + z$

Answer

The expression representing the product of the three integers,

x y z

, must always be an even integer.

The correct answer is the product of the three variables because logical parity analysis reveals that it is impossible for all three integers to be even, but in all valid cases at least one of the three variables must be an even integer (or zero). A product containing at least one even factor is always even.

Step-by-Step Solution

Express

z

in terms of

x

and

y

using the given sum equality.

z = -(x + y)

Since

x + y + z = 0

, substituting

z = -(x + y)

allows evaluation of the given parity condition in terms of

x

and

y

only.

Substitute

z = -(x + y)

into the given expression

x^3 y - z

and analyze its parity.

x^3 y - z = x^3 y + x + y

. Since

x^3

and

x

have the same parity for any integer

x

x^3 y + x + y

has the same parity as

x y + x + y

The parity of an integer raised to a positive integer power is identical to the parity of the base integer.

Evaluate the four possible parity combinations for

(x, y)

If both

x

and

y

are even, then

x y + x + y = \text{even} + \text{even} + \text{even} = \text{even}

, which contradicts the condition that

x^3 y - z

is odd. Thus,

x

and

y

cannot both be even.

Eliminating the case where both

x

and

y

are even implies that at least one of

x

y

must be odd.

Determine the parity of

x y z

in all remaining valid cases.

In Case 1 (

x

odd,

y

odd),

z = -(x + y) = -(\text{odd} + \text{odd}) = \text{even}

, so

z

is even and

x y z

is even. In Case 2 (

x

even,

y

odd),

x

is even so

x y z

is even. In Case 3 (

x

odd,

y

even),

y

is even so

x y z

is even. Therefore,

x y z

is even in every valid case.

The product of integers is even whenever at least one factor in the product is an even integer (including zero).

Key Concept

Parity rules of addition, multiplication, exponentiation, and the property that zero is an even integer.

Estimated Time:2m 0s

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