If k is a positive real number, is k an integer? (1) \sqrt{k} is an integer. (2) k^2 - 4k … — GMAT

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

(1) $\sqrt{k}$ is an integer.
(2) $k^2 - 4k$ is an integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
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.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (1) alone is sufficient because if the square root of

k

is an integer

m

, then

k = m^2

, which is guaranteed to be an integer. Statement (2) alone is not sufficient because

k^2 - 4k

can equal an integer when

k = 4

(an integer) or when

k = 2 + \sqrt{5}

(a non-integer). Therefore, only Statement (1) is sufficient.

Evaluate Statement (1) independently.

\sqrt{k} = m

where

m

is an integer, then

k = m^2

. Since

m

is an integer,

m^2

must also be an integer. Thus,

k

is definitely an integer (Yes).

The square of any integer is always an integer, so Statement (1) is sufficient.

Evaluate Statement (2) independently.

Let

k^2 - 4k = n

, where

n

is an integer. If

n = 0

, then

k(k - 4) = 0 \implies k = 4

(an integer). If

n = 1

, then

k^2 - 4k - 1 = 0 \implies k = 2 + \sqrt{5}

(not an integer).

Since

k

can be either an integer or a non-integer, Statement (2) does not provide a definitive Yes/No answer and is not sufficient.

Algebraic definitions of integers versus real numbers in Data Sufficiency

Estimated Time:2m 0s

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