If a, b, and c are integers such that a^2 + 3b is an odd integer, b^2 + 5c is an even inte… — GMAT

If $a$ , $b$ , and $c$ are integers such that $a^2 + 3b$ is an odd integer, $b^2 + 5c$ is an even integer, and $c^2 + 7a$ is an odd integer, which of the following expressions MUST be an even integer?

$a(b + c)$ Answer
B
$a + b + c$
C
$a + bc$
D
$ab + c$
E
$ac + b$

Answer

The expression

a(b + c)

MUST be an even integer.

The expression

a(b + c)

is guaranteed to be an even integer because in both valid parity configurations—whether

a

is odd with

b, c

even, or

a

is even with

b, c

odd—the sum

(b + c)

is always even, making the product

a(b + c)

even.

Step-by-Step Solution

Analyze the parity relationships from the given expressions.

Since

a^2

has the same parity as

a

3b

has the same parity as

b

b^2

has the same parity as

b

5c

has the same parity as

c

c^2

has the same parity as

c

, and

7a

has the same parity as

a

, the given conditions simplify to: (1)

a + b

is odd, (2)

b + c

is even, and (3)

c + a

is odd.

Multiplying an integer by an odd constant or raising an integer to a positive integer power does not change its parity.

Determine the valid parity configurations for

a

b

, and

c

Condition (1) implies

a

and

b

have opposite parities. Condition (2) implies

b

and

c

have the same parity. Condition (3) implies

c

and

a

have opposite parities. This yields two valid cases: Case 1:

b

and

c

are even, while

a

is odd. Case 2:

b

and

c

are odd, while

a

is even.

A sum of two integers is odd when they have opposite parities, and even when they have the same parity.

Evaluate the target expression

a(b + c)

across both valid cases.

In Case 1 (

a

odd,

b

even,

c

even):

b + c

is even, so

a(b + c) = \text{odd} \times \text{even} = \text{even}

. In Case 2 (

a

even,

b

odd,

c

odd):

b + c

is even, so

a(b + c) = \text{even} \times \text{even} = \text{even}

Any integer multiplied by an even integer yields an even integer.

Key Concept

Systematic parity deduction across multiple constraints and testing expressions under all valid parity configurations.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic