If m and n are positive real numbers, is m an integer? (1) m^2 n is an integer and m n^2 i… — GMAT

If $m$ and $n$ are positive real numbers, is $m$ an integer?

(1) $m^2 n$ is an integer and $m n^2$ is an integer.
(2) $m^3 n$ is an integer and $n$ 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.Answer

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.

Statements (1) and (2) together are not sufficient because setting

m = \frac{1}{2}

and

n = 8

satisfies all given conditions (

m^2 n = 2

m n^2 = 32

m^3 n = 1

n = 8

) while yielding a 'No' to whether

m

is an integer. Meanwhile,

m = 1

and

n = 8

also satisfies all conditions and yields a 'Yes'. Therefore, the two statements combined do not uniquely determine if

m

is an integer.

Step-by-Step Solution

Evaluate Statement (1) independently.

Statement (1) is insufficient.

m = 1

and

n = 2

, then

m^2 n = 2

and

m n^2 = 4

(both integers), and

m = 1

is an integer. However, if

m = \sqrt[3]{2}

and

n = \sqrt[3]{4}

, then

m^2 n = 2

and

m n^2 = 2

(both integers), but

m = \sqrt[3]{2}

is not an integer.

Evaluate Statement (2) independently.

Statement (2) is insufficient.

m = 1

and

n = 1

, then

m^3 n = 1

and

n = 1

(both integers), and

m = 1

is an integer. However, if

m = \frac{1}{2}

and

n = 8

, then

n = 8

is an integer and

m^3 n = (\frac{1}{8})(8) = 1

is an integer, but

m = \frac{1}{2}

is not an integer.

Evaluate Statements (1) and (2) together.

Statements (1) and (2) together are insufficient.

Test the candidate case

m = \frac{1}{2}

and

n = 8

against both statements simultaneously:
- Under Statement (1):

m^2 n = (\frac{1}{4})(8) = 2

(integer) and

m n^2 = (\frac{1}{2})(64) = 32

(integer).
- Under Statement (2):

n = 8

(integer) and

m^3 n = (\frac{1}{8})(8) = 1

(integer).
Here, all conditions from both statements are satisfied, but

m = \frac{1}{2}

is NOT an integer. Conversely,

m = 1

and

n = 8

also satisfies both statements and

m = 1

IS an integer. Thus, a definitive Yes/No answer cannot be determined.

Key Concept

Implicit real number constraints versus integer constraints in Data Sufficiency polynomial expressions.

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