Let x and y be non-zero integers that satisfy all of the following conditions: I. (-x)^{y … — GRE General Test

Let $x$ and $y$ be non-zero integers that satisfy all of the following conditions:
I. $(-x)^{y + 1} < 0$
II. $x^2 y + x$ is odd
III. $x < y$

Which of the following statements MUST be true? Select all that apply.

$y^x$ is an even integerAnswer
$x^y - y$ is an odd integerAnswer
C
$-x < -y$
$(x + y)^2$ is an odd integerAnswer
E
$(-x)^y < 0$

Answer

The statements that must be true are '

y^x

is an even integer', '

x^y - y

is an odd integer', and '

(x + y)^2

is an odd integer'.

From the given conditions, we deduce that

x

is a positive odd integer and

y

is a positive even integer with

x < y

. Therefore:
- Raising the positive even integer

y

to the positive power

x

yields an even integer (

y^x

is even).
- Raising the odd integer

x

to the positive power

y

yields an odd integer, and subtracting the even integer

y

leaves an odd integer (

x^y - y

is odd).
- The sum of odd

x

and even

y

is odd, and squaring an odd integer yields an odd integer (

(x + y)^2

is odd).

Step-by-Step Solution

Analyze Condition I:

(-x)^{y + 1} < 0

x > 0

(a positive integer) and

y

is an even integer.

For a number raised to an integer power to be strictly negative, the base must be negative and the exponent must be odd. Hence,

-x < 0 \implies x > 0

, and

y + 1

is odd

\implies y

is even.

Analyze Condition II:

x^2 y + x

is odd

x

is an odd integer.

Factor the expression as

x(xy + 1)

. A product of two integers is odd if and only if both factors are odd. Thus,

x

must be odd. (Additionally,

xy + 1

must be odd

\implies xy

is even, which holds since

y

is even).

Analyze Condition III:

x < y

y

is a positive even integer.

Since

x

is a positive odd integer (

x \ge 1

) and

x < y

y

must also be a positive integer (

y \ge 2

Evaluate the statements based on the derived properties (

x

is positive odd,

y

is positive even,

x < y

)

The statements '

y^x

is an even integer', '

x^y - y

is an odd integer', and '

(x + y)^2

is an odd integer' are guaranteed to be true.

\text{even}^{\text{positive odd}} = \text{even}

. 2)

\text{odd}^{\text{positive even}} - \text{even} = \text{odd} - \text{even} = \text{odd}

. 3)

(\text{odd} + \text{even})^2 = \text{odd}^2 = \text{odd}

Key Concept

Deducing parity and signs of variables using exponent rules and arithmetic properties of even and odd numbers.

Estimated Time:2m 0s

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