Is Positive Integer x Divisible By 24? GMAT Data Sufficiency

Question: Is positive integer x divisible by 24?

(1) √x is divisible by 4.
(2) x^2 is not divisible by 9.

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.

“Is positive integer x divisible by 24?” is a topic of the GMAT Quantitative reasoning section of GMAT. The questions of GMAT Data Sufficiency emerge with a problem statement along with two factual statements. This specific GMAT data sufficiency question tests the candidates’ talents in apprehending quantitative problems and solving them with mathematical calculations. The clever accents of words are one the toughest portion of these types of questions that candidates usually miss. The GMAT Quant section contains 31 MCQs and the time duration is 62 minutes. GMAT data sufficiency includes 15 questions out of these 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

The problem statement asks to find out if the positive integer x is divisible by 24.
To solve this question, it is required to find the prime factor of 24.
Therefore, we can write 24 as 2^3 * 3^1.

If x is divisible by 24 then the value of x should have at least all these prime factors.

1. The statement states that √x is divisible by 4.
   Therefore squaring both sides we can say, x is divisible by 16.
   Thus we can deduce that x= k*16 where the value of k is constant.
   Now, the number 16 can be written as 2^4 which is greater than 2^3.
   Therefore, if “k” contains at least a 3, then x is divisible by 24. But if “k” does not contain a 3, x is not divisible by 24.
   Thus we cannot assure from this statement whether x is divisible by 24 or not.
   Hence, the statement alone is not sufficient.
2. The statement cites that x^2 is not divisible by 9.
   Therefore, by doing the square root on both sides, we get,
   x is not divisible by 3.
   This means that the value of x does not contain 3 as one of its prime factors.
   However, to make x divisible by 24, the value of x must contain 3^1 i.e. one 3 as its prime factor.
   Therefore, we can infer that x is not divisible by 24 as it lacks the number 3 as one of its prime factors.
   Hence the statement alone is sufficient to find out whether x is divisible by 24.

Correct Answer: (B)

Approach Solution 2:

The problem statement asks to find out if the positive integer x is divisible by 24.
The question can be answered in the quick and easiest way.
That is we can proceed with the prime factorisation of 24.
Prime factorization of 24= 2^3* 3

1. The statement indicates that √x is divisible by 4.
   This means the value of x contains 4 2s (4* 4), which makes the number divisible by 8.
   The number 3 is nowhere cited in this equation. Hence the statement alone is not sufficient to find out if the positive integer x is divisible by 24.
2. The statement notes that x^2 is not divisible by 9.
   If the factor of x is 3 then x^2 would be divisible by 9 (3*3).
   Thus we can infer that x cannot be divisible by 24 since x does not have 3 as its factor. Hence the statement alone is sufficient.

Correct Answer: (B)

Approach Solution 3:

The problem statement questions to find out whether the positive integer x is divisible by 24 or not.
In order to find x divisible by 24, we need to focus on the factors of 24.
Therefore 24 = 3 ∗ 8 = 3* 2 * 2* 2 = 3^1* 2^3
Therefore, we can infer that any number divided by 24 must have one 3 and three 2’s.

1. The statement states that √x is divisible by 4 = √x/4
   Squaring both the numerator and denominator, we get,
   (√x/4)^2= x/16 = x/2*2*2*2 = x/24
   This means x= k*x is divisible by 2^3
   But we cannot ensure that k has 3 as its factor to make x divisible by 24.
   Hence, the statement alone is not sufficient.
2. The statement notes that x^2 is not divisible by 9.
   Since x^2 is not divisible by 9, x will not be divided by 3.
   In order to find x divisible by 24, one prime number 3 must be needed.
   Since x cannot be divided by 3, therefore x is also not divisible by 24.
   Thus the statement alone is sufficient to find the answer to this question.

Correct Answer: (B)

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