Even-Odd Properties and Sign Rules

22 questions

All types22 Multiple choice11 Multiple select6 Numeric5

Let $r$ and $s$ be integers such that $(-1)^{r+s} = -1$ and $r^2 s + r$ is an odd integer. Which of the following statements must be true? Select all such statements.

Select all that apply

r - s

is an odd integer.

r + 2s

is an even integer.

r^2 + s^2

is an odd integer.

s(r + 1)

is an odd integer.

r^2 s + s

is an even integer.

Show answer & explanation

Answer: $r - s$ is an odd integer.; $r^2 + s^2$ is an odd integer.; $r^2 s + s$ is an even integer.

Answer

The statements that must be true are '

r - s

is an odd integer', '

r^2 + s^2

is an odd integer', and '

r^2 s + s

is an even integer'.

From

(-1)^{r+s} = -1

, the sum

r+s

must be odd, meaning

r

and

s

have opposite parity. Factoring

r^2 s + r

yields

r(rs + 1) = \text{odd}

, which requires both

r

and

rs + 1

to be odd. Hence,

r

is odd, which forces

s

to be even. Testing the options shows that subtracting an even number from an odd number gives an odd number, adding the squares of an odd and an even number gives an odd number, and multiplying any integer by the even number

s

gives an even number.

Step-by-Step Solution

Determine the parity of

r + s

from

(-1)^{r+s} = -1

r + s

is an odd integer

For

(-1)^k = -1

, the exponent

k

must be an odd integer. Therefore,

r + s

is odd, which implies that one variable is even and the other is odd.

Analyze the given expression

r^2 s + r

r

is odd and

s

is even

Factor

r^2 s + r

r(rs + 1)

. For the product of two integers to be odd, both factors must be odd. Thus,

r

must be odd. Since

r + s

is odd and

r

is odd,

s

must be even. (Verification: if

s

is even and

r

is odd,

rs + 1

is even + 1 = odd, so

r(rs + 1)

is odd

\times

odd = odd).

Evaluate each given statement using

r = \text{odd}

and

s = \text{even}

Statements '

r - s

is an odd integer', '

r^2 + s^2

is an odd integer', and '

r^2 s + s

is an even integer' are true.

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

(True).
2)

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

(False for even).
3)

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

(True).
4)

(\text{even})(\text{odd} + 1) = \text{even} \times \text{even} = \text{even}

(False for odd).
5)

r^2 s + s = s(r^2 + 1) = \text{even} \times \text{even} = \text{even}

(True).

Key Concept

Parity rules for integer addition, multiplication, and exponents

Question 22Question →

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.

Select all that apply

x

is a negative integer

xy

is an even integer

x^2 + y^2

is an odd integer

(-x)^y

is guaranteed to be a positive integer

x < y

, then

-x < -y

Show answer & explanation

Answer: $x$ is a negative integer; $xy$ is an even integer; $x^2 + y^2$ is an odd integer

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

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Even-Odd Properties and Sign Rules

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes