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All Quantitative Aptitude (QA) 1 / 33

Each of the following questions consists of a question and two statements, numbered (1) and (2). You must decide whether the data provided in the statements are sufficient to answer the question.

Question: Is the integer n divisible by 36?

(1) n is divisible by 9.

(2) n is divisible by 4.

(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 (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Correct Answer: (C)

1. Analysis of the Question: For an integer n to be divisible by 36, it must be simultaneously divisible by the coprime factors of 36. Since 36 = 4 * 9 and the greatest common divisor of 4 and 9 is 1, a number divisible by both 4 and 9 is necessarily divisible by 36.

2. Evaluating Statement (1): Statement (1) says n is divisible by 9. This means n could be 9, 18, 27, 36, 45, etc. Some of these are divisible by 36 (e.g., 36), while others are not (e.g., 9). Therefore, statement (1) alone is not sufficient.

3. Evaluating Statement (2): Statement (2) says n is divisible by 4. This means n could be 4, 8, 12, 16, 20, 24, 28, 32, 36, etc. Some are divisible by 36, while others are not. Therefore, statement (2) alone is not sufficient.

4. Combining Both Statements: When we use both statements, we know n is divisible by 9 (from statement 1) and n is divisible by 4 (from statement 2). Since 4 and 9 are coprime (they share no factors other than 1), any number divisible by both must be divisible by their product, 36. This provides a unique "yes" to the question.

5. Logical Verification: The Least Common Multiple (LCM) of 4 and 9 is 36. If a number is a multiple of both 4 and 9, it must be a multiple of their LCM. Since n is a multiple of 36, it is by definition divisible by 36.

Test Prep Tip: In Data Sufficiency questions involving divisibility, always break the divisor into its prime factors or coprime pairs. A common trap is using non-coprime factors; for example, if the question asked about divisibility by 24, knowing n is divisible by 4 and 6 would NOT be sufficient because their LCM is 12, not 24. Always check for coprimality.

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