If n is a positive real number, is n an integer? (1) n + \frac{6}{n} is an integer. (2) n^… — GMAT

If $n$ is a positive real number, is $n$ an integer?

(1) $n + \frac{6}{n}$ is an integer.
(2) $n^2 + \frac{36}{n^2}$ 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.
C
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.Cevap

Cevap

Statements (1) and (2) TOGETHER are NOT sufficient to determine whether

n

is an integer.

The correct answer identifies that neither statement alone is sufficient, and combining them offers no additional constraint because the second statement is algebraically derived by squaring the first. Since non-integer real numbers such as

n = 3 + \sqrt{3}

satisfy both statements, the statements together remain insufficient.

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Evaluate Statement (1) by setting

n + \frac{6}{n} = k

, where

k

is an integer.

Multiplying by

n

gives

n^2 - kn + 6 = 0

, which yields

n = \frac{k \pm \sqrt{k^2 - 24}}{2}

. For

n

to be a real number, we need

k^2 \ge 24

, so

k \ge 5

We must test whether all positive real solutions for

n

under this condition are integers.

Test specific integer values for

k

in Statement (1).

k = 5

, then

n = \frac{5 \pm 1}{2}

, giving

n = 3

n = 2

(both are integers, yielding a YES answer). However, if

k = 6

, then

n = \frac{6 \pm \sqrt{12}}{2} = 3 \pm \sqrt{3}

, which is a positive real number but NOT an integer (yielding a NO answer).

Since Statement (1) permits both integer and non-integer values for

n

, Statement (1) ALONE is NOT sufficient.

Evaluate Statement (2) independently and analyze its algebraic relationship to Statement (1).

n + \frac{6}{n} = k

, then squaring both sides gives

\left(n + \frac{6}{n}\right)^2 = n^2 + 12 + \frac{36}{n^2} = k^2

, which implies

n^2 + \frac{36}{n^2} = k^2 - 12

. If

k

is an integer,

k^2 - 12

is automatically an integer.

This shows Statement (2) is logically and algebraically implied by Statement (1).

Evaluate Statements (1) and (2) together.

Because Statement (2) provides zero new information beyond Statement (1), combining them does not eliminate the non-integer solution

n = 3 + \sqrt{3}

. Thus, the question cannot be answered definitively.

Redundant statements cannot resolve insufficiency present in the individual statements.

Anahtar Kavram

Number Properties and Integer Constraints in Data Sufficiency

Tahmini Süre:2m 30s

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