If k and m are integers such that k < 0 < m, (-1)^k = -1, and (-1)^m = 1, which of the fol… — GRE General Test

If $k$ and $m$ are integers such that $k < 0 < m$ , $(-1)^k = -1$ , and $(-1)^m = 1$ , which of the following expressions MUST be a negative odd integer?

$k - m$ Answer
B
$k + m$
C
$k m$
D
$m - k$
E
$k^2 - m$

Answer

The expression

k - m

MUST be a negative odd integer.

Given

k < 0 < m

k

is negative and

m

is positive. The relation

(-1)^k = -1

shows

k

is odd, while

(-1)^m = 1

shows

m

is even. Subtracting a positive even integer

m

from a negative odd integer

k

yields

k - m

, which must be less than 0 (negative) and odd (odd minus even).

Step-by-Step Solution

Determine the parity and sign of

k

k

is a negative odd integer.

k < 0

specifies that

k

is negative, and

(-1)^k = -1

implies that the exponent

k

must be odd.

Determine the parity and sign of

m

m

is a positive even integer.

m > 0

specifies that

m

is positive, and

(-1)^m = 1

implies that the exponent

m

must be even.

Evaluate the sign and parity of

k - m

k - m

is strictly negative and odd.

Since

k < 0

and

m > 0

k - m = k + (-m) < 0

. By parity rules,

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

Key Concept

Even-Odd Properties and Sign Rules

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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