If x is an odd integer and y is an even integer, which of the following expressions must r… — GRE General Test

If $x$ is an odd integer and $y$ is an even integer, which of the following expressions must result in an even integer? Select all that apply.

The expressions that must be even are

xy + y

and

(x + 1)(y + 1)

The expression

xy + y

is guaranteed to be even because

xy

is even (product of odd and even) and adding another even integer

y

produces an even sum. The expression

(x + 1)(y + 1)

is also guaranteed to be even because adding 1 to an odd integer

x

creates an even integer

x + 1

, and multiplying an even number by any integer always yields an even product.

Analyze fundamental parity rules for addition and multiplication of integers.

Recall that

\text{odd} \times \text{even} = \text{even}

\text{odd} \times \text{odd} = \text{odd}

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

, and

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

Parity rules govern the even/odd behavior of combined expressions.

Evaluate the expression

xy + y

Since

x

is odd and

y

is even,

xy

is even. Then

\text{even} + y\,(\text{even}) = \text{even}

Adding two even terms always results in an even number.

Evaluate the expression

(x + 1)(y + 1)

Since

x

is odd,

x + 1

is even. The product of an even integer and any integer (

y + 1

) is always even.

An even factor guarantees an even product.

Even and Odd Parity Rules under Arithmetic Operations

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