Let x and y be non-zero integers such that x^5 y^2 < 0 and x + y is an odd integer. Which … — GRE General Test

Let $x$ and $y$ be non-zero integers such that $x^5 y^2 < 0$ and $x + y$ is an odd integer. Which of the following statements MUST be true? Select all that apply.

$x$ is a negative integerAnswer
$xy$ is an even integerAnswer
$x^2 + y^2$ is an odd integerAnswer
D
$(-x)^y$ is guaranteed to be a positive integer
E
If $x < y$ , then $-x < -y$

Answer

The statements '

x

is a negative integer', '

xy

is an even integer', and '

x^2 + y^2

is an odd integer' MUST be true.

The statement specifying that '

x

is a negative integer' is true because

y^2

is strictly positive for any non-zero integer

y

, forcing

x^5 < 0

and thus

x < 0

. The statement '

xy

is an even integer' is true because

x + y

being odd requires one variable to be even and the other to be odd, making their product even. The statement '

x^2 + y^2

is an odd integer' is true because the square of an even number is even and the square of an odd number is odd, and their sum is always odd.

Step-by-Step Solution

Determine the sign of

x

from the given inequality

x^5 y^2 < 0

x

must be negative (

x < 0

Since

y

is a non-zero integer,

y^2 > 0

. Dividing the inequality by

y^2

gives

x^5 < 0

, which means

x

must be negative.

Analyze the parity (even/odd nature) of

x

and

y

using

x + y

is odd.

One of

x, y

is even and the other is odd.

The sum of two integers is odd if and only if one integer is even and the other is odd.

Evaluate the statement '

xy

is an even integer'.

The product

xy

is always even.

The product of an even integer and an odd integer is always even.

Evaluate the statement '

x^2 + y^2

is an odd integer'.

The sum of squares

x^2 + y^2

is always odd.

Squaring an even integer yields an even integer, and squaring an odd integer yields an odd integer. Adding an even number and an odd number yields an odd number.

Key Concept

Even-Odd Parity Rules and Exponent Sign Properties

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

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