If k is a positive integer, is k divisible by 6? (1) k is a multiple of 2. (2) k is a mult… — GMAT

If $k$ is a positive integer, is $k$ divisible by 6?

(1) $k$ is a multiple of 2.
(2) $k$ is a multiple of 3.

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.Answer
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Both statements together are sufficient, but neither statement alone is sufficient.

To determine whether an integer is divisible by 6, it must be divisible by both prime factors of 6, which are 2 and 3. Statement (1) indicates

k

is divisible by 2, and Statement (2) indicates

k

is divisible by 3. Neither statement alone is sufficient because a number can be a multiple of 2 without being a multiple of 3 (e.g., 2, 4) or a multiple of 3 without being a multiple of 2 (e.g., 3, 9). Taking both statements together ensures

k

is a multiple of both 2 and 3, which means

k

is a multiple of 6. Therefore, both statements together are sufficient.

Step-by-Step Solution

Rephrase the question target.

For a positive integer

k

to be divisible by 6, it must be divisible by both 2 and 3 because

\text{lcm}(2,3) = 6

6 is the product of the prime numbers 2 and 3.

Evaluate Statement (1) independently.

k = 2

, it is divisible by 2 but not by 6 (No). If

k = 6

, it is divisible by 2 and by 6 (Yes). Insufficient.

Statement (1) alone allows multiple outcomes for divisibility by 6.

Evaluate Statement (2) independently.

k = 3

, it is divisible by 3 but not by 6 (No). If

k = 6

, it is divisible by 3 and by 6 (Yes). Insufficient.

Statement (2) alone allows multiple outcomes for divisibility by 6.

Combine Statement (1) and Statement (2).

Since

k

is divisible by 2 and divisible by 3,

k

must be a multiple of

\text{lcm}(2,3) = 6

. Thus,

k

is definitely divisible by 6 (Definitive Yes).

Both conditions combined uniquely answer the question with a definitive Yes.

Key Concept

Divisibility Rules and Prime Factorization

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