If x, y, and z are integers such that (x + 1)(y + 2) is an odd integer and y(z + 3) is an … — GMAT

If $x$ , $y$ , and $z$ are integers such that $(x + 1)(y + 2)$ is an odd integer and $y(z + 3)$ is an even integer, which of the following expressions MUST be an even integer?

$x + y + z$ Answer
B
$xy + z$
C
$(y + z)^2 + 1$
D
$xz + y$
E
$(x + 1)yz$

Answer

The expression

x + y + z

MUST be an even integer.

The condition

(x + 1)(y + 2) = \text{odd}

implies both

(x + 1)

and

(y + 2)

are odd, so

x

is even and

y

is odd. Then

y(z + 3) = \text{even}

with

y

being odd requires

(z + 3)

to be even, so

z

is odd. Summing an even integer

x

and two odd integers

y

and

z

yields

\text{even} + \text{odd} + \text{odd} = \text{even}

. Thus, the expression representing the sum of

x

y

, and

z

must always be even.

Step-by-Step Solution

Determine the parity of

x

and

y

from the first given condition.

x

is even and

y

is odd.

The product

(x + 1)(y + 2)

is odd, which means both factors

(x + 1)

and

(y + 2)

must be odd. If

x + 1

is odd, then

x

is even. If

y + 2

is odd, then

y

is odd.

Determine the parity of

z

using the parity of

y

and the second given condition.

z

is odd.

The product

y(z + 3)

is even. Since

y

is odd,

(z + 3)

must be even for the product to be even. If

z + 3

is even, then

z

must be odd.

Evaluate the parity of

x + y + z

x + y + z

is even.

Substituting the parities gives

\text{even} + \text{odd} + \text{odd} = \text{even} + \text{even} = \text{even}

Key Concept

Parity rules for addition and multiplication of integers

Estimated Time:2m 0s

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