If x and y are positive real numbers, is the product xy an odd integer? (1) x^2y is an odd… — GMAT

If $x$ and $y$ are positive real numbers, is the product $xy$ an odd integer?

(1) $x^2y$ is an odd integer.
(2) $xy^2$ is an odd 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.

The correct option is the choice stating that Statements (1) and (2) together are NOT sufficient. Since

x

and

y

are positive real numbers, setting

x = y = \sqrt[3]{3}

satisfies both

x^2y = 3

and

xy^2 = 3

(both odd integers), but gives

xy = \sqrt[3]{9}

, which is not an integer. Conversely, setting

x = 1/\sqrt[3]{3}

and

y = \sqrt[3]{9}

yields

x^2y = 3

and

xy^2 = 9

(both odd integers), giving

xy = 3

, which is an odd integer. Because both a 'No' and a 'Yes' answer are possible, the information remains insufficient even when both statements are combined.

Step-by-Step Solution

Analyze the stem constraints and target question.

The target asks whether

xy

is an odd integer, given that

x

and

y

are positive real numbers (not necessarily integers).

Recognizing that

x

and

y

are not restricted to integers is essential to avoid invalid integer assumption traps.

Evaluate Statement (1) independently.

Statement (1) gives

x^2y = k

for some odd integer

k

. If

x = \sqrt{3}

and

y = 1

, then

x^2y = 3

(an odd integer), but

xy = \sqrt{3}

(not an integer). If

x = 3

and

y = 1

, then

x^2y = 9

(an odd integer) and

xy = 3

(an odd integer). Thus, Statement (1) alone is NOT sufficient.

Testing non-integer values demonstrates that

xy

may or may not be an integer.

Evaluate Statement (2) independently.

Statement (2) gives

xy^2 = m

for some odd integer

m

. If

x = 1

and

y = \sqrt{3}

, then

xy^2 = 3

(an odd integer), but

xy = \sqrt{3}

(not an integer). If

x = 1

and

y = 3

, then

xy^2 = 9

(an odd integer) and

xy = 3

(an odd integer). Thus, Statement (2) alone is NOT sufficient.

Symmetry with Statement (1) confirms that Statement (2) alone produces variable outcomes.

Evaluate Statements (1) and (2) combined.

Multiplying the two expressions gives

(x^2y)(xy^2) = x^3y^3 = (xy)^3 = km

. Since

k

and

m

are odd integers, their product

km

is also an odd integer, say

P

. Thus,

(xy)^3 = P \implies xy = \sqrt[3]{P}

. If

k = 3

and

m = 3

, then

P = 9

, so

xy = \sqrt[3]{9}

(not an integer). If

k = 3

and

m = 9

, then

P = 27

, so

xy = \sqrt[3]{27} = 3

(an odd integer). Because

xy

can be either an odd integer or a non-integer, the combined statements are NOT sufficient.

Combining the statements yields

(xy)^3

equal to an odd integer, which does not guarantee that

xy

itself is an integer.

Key Concept

Implicit Real vs. Integer Constraints in Data Sufficiency

Estimated Time:2m 0s

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