Is the integer x odd? Statement (I): x^2 + 3x is an even integer. Statement (II): x + 5 is… — State PSC Exam

Is the integer $x$ odd?

Statement (I): $x^2 + 3x$ is an even integer.
Statement (II): $x + 5$ is an even integer.

Which of the following options is correct?

A
Statement (I) alone is sufficient, but Statement (II) alone is not sufficient.
Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.Answer
C
Either Statement (I) alone or Statement (II) alone is sufficient.
D
Neither Statement (I) nor Statement (II) is sufficient.
E
Both Statement (I) and Statement (II) together are sufficient, but neither statement alone is sufficient.

Statement (II) alone is sufficient, but Statement (I) alone is not sufficient.

Evaluating Statement (I):

x^2 + 3x = x(x+3)

. The product of any integer

x

and

(x+3)

is always even because one of the two numbers is always even. Thus, Statement (I) provides no specific information about whether

x

is odd or even, making it insufficient. Evaluating Statement (II):

x + 5 = \text{even}

. Subtracting the odd integer

5

from an even integer yields an odd integer, so

x

must be odd. Hence, Statement (II) alone is sufficient to answer the question definitively.

Evaluate Statement (I) individually.

Rewrite the expression as

x^2 + 3x = x(x + 3)

. If

x

is even, then

x(x + 3) = \text{even} \times \text{odd} = \text{even}

. If

x

is odd, then

x(x + 3) = \text{odd} \times \text{even} = \text{even}

. Thus,

x^2 + 3x

is always even regardless of whether

x

is odd or even.

Since the statement holds true for all integers

x

, it cannot determine whether

x

is odd. Therefore, Statement (I) alone is NOT sufficient.

Evaluate Statement (II) individually.

The statement gives that

x + 5

is an even integer. Since

5

is an odd integer,

\text{Odd} + \text{Odd} = \text{Even}

, which implies

x

must be an odd integer.

This yields a definitive 'Yes' answer to the question 'Is

x

odd?'. Therefore, Statement (II) alone IS sufficient.

Conclude the data sufficiency evaluation.

Statement (II) alone is sufficient to answer the question, but Statement (I) alone is not sufficient.

Each statement was evaluated independently first, rendering statement combination unnecessary.

Data Sufficiency Parity Analysis

Estimated Time:1m 30s

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