If k is a positive integer, is \frac{k^3 - 4k}{k + 2} an odd integer? (1) k^2 + 3k is an e… — GMAT

If $k$ is a positive integer, is $\frac{k^3 - 4k}{k + 2}$ an odd integer?

(1) $k^2 + 3k$ is an even integer.
(2) $3k + 1$ is an odd integer.

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Rephrasing the target question

\frac{k^3 - 4k}{k + 2}

by factoring the numerator gives

\frac{k(k-2)(k+2)}{k+2} = k(k-2)

. Because

k

and

k-2

share the same parity, their product

k(k-2)

is odd if and only if

k

is odd. Thus, the question simplifies to: 'Is

k

an odd integer?' Statement (1) tells us

k^2 + 3k = k(k+3)

is even, which is true for all integers

k

, so it provides no information about whether

k

is odd or even (NOT sufficient). Statement (2) tells us

3k + 1

is odd, which implies

3k

is even, and therefore

k

must be even. Since

k

is definitively even, the answer to 'Is

k

an odd integer?' is a definitive 'No'. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

Rephrase and simplify the question stem algebraically.

Since

k

is a positive integer,

k + 2 > 0

. Factoring the numerator gives

k^3 - 4k = k(k^2 - 4) = k(k - 2)(k + 2)

. Dividing by

k + 2

yields

\frac{k(k - 2)(k + 2)}{k + 2} = k(k - 2)

Simplifying the target expression reduces cognitive complexity before evaluating statements.

Determine the parity condition for

k(k - 2)

to be an odd integer.

The factors

k

and

k - 2

differ by 2, so they have the same parity. If

k

is even,

k(k - 2)

is (even)(even) = even. If

k

is odd,

k(k - 2)

is (odd)(odd) = odd. Thus, the target question 'Is

k(k - 2)

an odd integer?' simplifies to 'Is

k

an odd integer?'

Rephrasing the target question to 'Is

k

odd?' establishes a clear, minimal sufficiency criterion.

Evaluate Statement (1):

k^2 + 3k

is an even integer.

Factor

k^2 + 3k

k(k + 3)

. The terms

k

and

k + 3

differ by 3, meaning one is always even and the other is odd. Their product is ALWAYS even regardless of whether

k

is odd or even. Therefore, Statement (1) provides no information about whether

k

is odd. NOT sufficient.

A statement that holds true for all integers provides no distinguishing information about

k

Evaluate Statement (2):

3k + 1

is an odd integer.

3k + 1

is odd, then

3k

must be even. Since 3 is odd,

3k

is even if and only if

k

is even. Hence,

k

is definitely even. This gives a definitive 'No' to the rephrased question 'Is

k

an odd integer?'. A definitive 'No' means the statement IS sufficient.

In Data Sufficiency Yes/No questions, a statement that yields a conclusive 'No' is sufficient.

Key Concept

Data Sufficiency Question Stem Simplification and Parity Analysis

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

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