Let a, b, and c be non-zero integers such that \frac{a}{b} < 0, a^3 b c > 0, and a + b is … — GRE General Test

Let $a$ , $b$ , and $c$ be non-zero integers such that $\frac{a}{b} < 0$ , $a^3 b c > 0$ , and $a + b$ is an odd integer. Which of the following statements must be true? Select all such statements.

$c < 0$ Answer
$a b$ is an even integerAnswer
C
$a c > 0$
D
$a^2 + c$ is an even integer
E
$b + c < 0$

Answer

The statements '

c < 0

' and '

a b

is an even integer' must be true.

The statement '

c < 0

' must be true because

\frac{a}{b} < 0

forces

a b < 0

, and since

a^3 b c = a^2 (a b) c

with

a^2 > 0

c

must be negative to yield a positive product. The statement '

a b

is an even integer' must be true because an odd sum

a + b

requires one variable to be even and the other to be odd, making their product even.

Step-by-Step Solution

Determine the sign relationship between

a

and

b

a

and

b

have opposite signs, so

a b < 0

The quotient

\frac{a}{b} < 0

implies the numerator and denominator have different signs.

Determine the sign of

c

using

a^3 b c > 0

c < 0

Rewrite

a^3 b c

a^2 \cdot (a b) \cdot c

. Since

a \neq 0

a^2 > 0

. Since

a b < 0

, the product

a^2 (a b) < 0

. For the entire product

a^2 (a b) c

to be positive,

c

must be negative.

Analyze the parity of

a

and

b

from

a + b

being odd.

One of

a

b

is even and the other is odd, so

a b

must be even.

An odd sum of two integers requires one even and one odd addend. The product of an even integer and any integer is always even.

Test the remaining options for counterexamples.

The statements '

a c > 0

', '

a^2 + c

is an even integer', and '

b + c < 0

' can be false under valid assignments.

For example, if

a = 2

b = -1

, and

c = -3

, then

\frac{a}{b} = -2 < 0

a^3 b c = 8(-1)(-3) = 24 > 0

, and

a + b = 1

(odd). Here,

a c = -6 < 0

a^2 + c = 4 - 3 = 1

(odd), and

b + c = -4 < 0

, but setting

b = 5, a = -2, c = -1

gives

b + c = 4 > 0

Key Concept

Deducing sign and parity properties of integers

Estimated Time:1m 30s

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