Let r and s be integers such that (-1)^{r+s} = -1 and r^2 s + r is an odd integer. Which o… — GRE General Test

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.

$r - s$ is an odd integer.Answer
B
$r + 2s$ is an even integer.
$r^2 + s^2$ is an odd integer.Answer
D
$s(r + 1)$ is an odd integer.
$r^2 s + s$ is an even integer.Answer

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

Answer

Step-by-Step Solution

Key Concept

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