GMAT Data Sufficiency - If x is an integer is \(\frac{x}{12}\) an integer?

Question: If x is an integer is \(\frac{x}{12}\) an integer?

1. x is a multiple of both 4 and 6
2. x is a multiple of both 8 and 10

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If x is an integer is \(\frac{x}{12}\) an integer?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Approach Solution 1

Firstly consider S (1), we have x is a multiple of both 4 and 6. Therefore x is a multiple of least common multiple of 4 and 6, so a multiple of 12, hence is \(\frac{x}{12}\) an integer.

Hence, this statement is sufficient.

Now consider S (2), we have x is a multiple of both 8 and 10. Therefore, x is a multiple of 8 and 10, so a multiple of 40. Now, if x = 40 then the answer is NO but if x = 120 then the answer is YES.

Hence, the statement is NOT sufficient.

Correct option: A

Approach Solution 2

Another way to look at this is by saying that for \(\frac{x}{12}\) to be an integer, it must be able to divide by BOTH 3 and 4.

Firstly consider S (1), we have both 3 and 4 are factors of 4 and 6

Hence this statement is SUFFICIENT.

Now consider S (2), we see that 4 is a factor of 8 but 3 is not a factor of neither 8 nor 10

Hence, this statement is NOT sufficient. Correct option: A

Correct option: A

Approach Solution 3

Firstly consider S (1), x is a multiple of both 4 and 6

Sufficient: x is a multiple of both 4 and 6 i.e., x is a multiple of LCM (4,6) = 12, so  \(\frac{x}{12}\)= integer.

Now consider S (2), x is a multiple of both 8 and 10

Insufficient: x is a multiple of both 8 and 18 i.e., x is multiple of LCM (8,10) = 40, so \(\frac{x}{12}\) is not necessarily an integer.

Correct option: A

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