If x is a positive real number, is x an integer? (1) x^3 is an integer. (2) x^5 is an inte… — GMAT

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

(1) $x^3$ is an integer.
(2) $x^5$ 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.Cevap
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

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

Evaluating each statement independently leaves open the possibility that

x

is an irrational root like

\sqrt[3]{2}

\sqrt[5]{2}

, making neither statement alone sufficient. Combining both statements allows us to express

x

\frac{(x^3)^2}{x^5}

. Because both

x^3

and

x^5

are integers,

x

must be a rational number. For any rational number whose cube is an integer, its denominator must equal 1, which proves conclusively that

x

is an integer. Thus, both statements together are sufficient.

Adım Adım Çözüm

Evaluate Statement (1) independently.

Statement (1) is insufficient.

x = 2

, then

x^3 = 8

(an integer), so

x

is an integer (Yes). However, if

x = \sqrt[3]{2}

, then

x^3 = 2

(an integer), but

x

is not an integer (No). Since both Yes and No answers are possible, Statement (1) alone is not sufficient.

Evaluate Statement (2) independently.

Statement (2) is insufficient.

x = 2

, then

x^5 = 32

(an integer), so

x

is an integer (Yes). However, if

x = \sqrt[5]{2}

, then

x^5 = 2

(an integer), but

x

is not an integer (No). Since both Yes and No answers are possible, Statement (2) alone is not sufficient.

Combine Statement (1) and Statement (2).

The combined statements are sufficient.

Notice that

\frac{(x^3)^2}{x^5} = \frac{x^6}{x^5} = x

. Since

x^3

is an integer,

(x^3)^2

is also an integer. Since

x^5

is an integer,

x = \frac{(x^3)^2}{x^5}

is the ratio of two integers, meaning

x

must be a rational number. If

x = \frac{a}{b}

in simplest form where

a

and

b

are positive integers with

\gcd(a,b)=1

, then

x^3 = \frac{a^3}{b^3}

being an integer implies

b^3 = 1

, so

b = 1

. Thus,

x = a

, which means

x

must be an integer. This yields a definitive 'Yes' answer.

Anahtar Kavram

Integer constraints versus real numbers in Data Sufficiency and quotient relationships of exponent powers

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