If k is a real number, is k an integer? (1) k^2 is an integer. (2) k^3 is an integer. — GMAT

If $k$ is a real number, is $k$ an integer?

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

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

The correct option states that both statements together are sufficient, but neither alone is sufficient. Statement (1) allows irrational square roots like

\sqrt{2}

, making it insufficient. Statement (2) allows irrational cube roots like

\sqrt[3]{2}

, making it insufficient. Combined, taking the ratio

\frac{k^3}{k^2} = k

shows that

k

is the quotient of two integers and therefore rational. A rational number whose square is an integer must be an integer itself, giving a definitive 'Yes'.

Step-by-Step Solution

Evaluate Statement (1):

k^2

is an integer.

k = 2

k^2 = 4

(an integer) and

k

is an integer (Yes). If

k = \sqrt{2}

k^2 = 2

(an integer), but

k

is not an integer (No). Statement (1) is NOT sufficient.

The question specifies

k

is a real number, so non-integer square roots must be tested.

Evaluate Statement (2):

k^3

is an integer.

k = 2

k^3 = 8

(an integer) and

k

is an integer (Yes). If

k = \sqrt[3]{2}

k^3 = 2

(an integer), but

k

is not an integer (No). Statement (2) is NOT sufficient.

Cube roots of non-cube integers yield irrational real numbers.

Evaluate Statement (1) and Statement (2) combined.

k = 0

k

is an integer (Yes). If

k \neq 0

, then

k = \frac{k^3}{k^2}

. Since

k^3

and

k^2

are both integers,

k

must be a rational number. Let

k = \frac{p}{q}

in simplest form where

p, q \in \mathbb{Z}

and

\gcd(p, q) = 1

. Since

k^2 = \frac{p^2}{q^2}

is an integer and

\gcd(p^2, q^2) = 1

q^2

must equal

1

, so

q = \pm 1

. Thus,

k

is an integer. Combined, the statements are SUFFICIENT.

Combining algebraic operations reveals that

k

is rational, and a rational number with an integer square must itself be an integer.

Key Concept

Data Sufficiency evaluation of number properties with real number constraints and rational number properties

Question

Answer

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