If a and b are integers such that a^2b + a is an odd integer, which of the following expre… — GMAT

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

The expression

a + b + 1

must be an even integer.

Factoring

a^2b + a

gives

a(ab + 1)

. For a product of two integers to be odd, both factors must be odd. Therefore,

a

is odd and

ab + 1

is odd. If

ab + 1

is odd, then

ab

must be even. Since

a

is odd,

b

must be even. Evaluating the expression

a + b + 1

a

(odd) +

b

(even) + 1 (odd) equals an even integer.

Factor the given algebraic expression to analyze its parity components.

a^2b + a = a(ab + 1)

Factoring allows us to analyze the parity of individual factors whose product is given as odd.

Determine the parity of

a

and the factor

(ab + 1)

Both

a

and

(ab + 1)

must be odd integers.

The product of two integers is odd if and only if both integer factors are odd.

Determine the parity of

b

using the parities established in Step 2.

Since

ab + 1

is odd,

ab

must be even. Because

a

is odd,

b

must be even (including zero).

An odd number multiplied by an even number yields an even number, and adding 1 converts it to an odd number.

Evaluate the parity of

a + b + 1

\text{odd} + \text{even} + 1 = \text{odd} + 1 = \text{even}

Adding two odd numbers (

a

and 1) together with an even number (

b

) results in an even integer.

Parity rules for integer addition and multiplication: Odd × Odd = Odd, Odd × Even = Even, Odd + Even = Odd, Odd + Odd = Even.

Question