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

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

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

Evaluating Statement (1) alone allows

x = \sqrt{2}

(not an integer) or

x = 2

(an integer), so Statement (1) is insufficient. Evaluating Statement (2) alone allows

x = \frac{1}{3}

(not an integer) or

x = 1

(an integer), so Statement (2) is insufficient. Combining both statements establishes that

x = \frac{m}{3}

for an integer

m

, and

x^2 = \frac{m^2}{9} = k

for an integer

k

. This requires

m^2 = 9k

, which implies

m

must be a multiple of 3. Hence

x

must be an integer, giving a definitive 'Yes' answer. Thus, both statements together are sufficient.

Step-by-Step Solution

Analyze the question stem constraint

x

is constrained to be a real number (

x \in \mathbb{R}

). The target question is a Yes/No question: 'Is

x

an integer?'

Without an explicit integer constraint in the stem, non-integer real numbers must be tested as counterexamples.

Evaluate Statement (1) alone

x = \sqrt{2}

, then

x^2 = 2

(an integer), but

x

is not an integer (Answer: No). If

x = 2

, then

x^2 = 4

(an integer), and

x

is an integer (Answer: Yes). Statement (1) is NOT sufficient.

Statement (1) allows both integer and non-integer values for

x

Evaluate Statement (2) alone

x = \frac{1}{3}

, then

3x = 1

(an integer), but

x

is not an integer (Answer: No). If

x = 1

, then

3x = 3

(an integer), and

x

is an integer (Answer: Yes). Statement (2) is NOT sufficient.

Statement (2) allows fractional values with a denominator of 3 as well as integers.

Evaluate Statements (1) and (2) together

From Statement (2),

x = \frac{m}{3}

for some integer

m

. Substituting into Statement (1) yields

x^2 = \left(\frac{m}{3}\right)^2 = \frac{m^2}{9} = k

, where

k

is an integer. Thus,

m^2 = 9k

. Since

9k

is a multiple of 9,

m^2

is divisible by 9, which means

m

must be a multiple of 3. Let

m = 3p

for some integer

p

. Then

x = \frac{3p}{3} = p

, which guarantees that

x

is an integer. The answer is a definitive 'Yes'. Statements (1) and (2) together are SUFFICIENT.

Combining both conditions restricts

x

to rational numbers whose square is an integer, forcing

x

to be an integer.

Key Concept

Number Properties and Integer Constraints in Data Sufficiency

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

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