Question

If \(x\) and \(y\) are two distinct positive integers and their product is \(54\), what is the value of \(x\)? Statements: (I) \(x\) is an odd integer. (II) \(x>y\)

• A. Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
• B. Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
• C. Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
• D. Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.

Hint:

In factor-based data sufficiency, list all factor pairs carefully and eliminate according to the given conditions.

Answer 1

The Correct Option is D

Concept: Given: \[ xy=54 \] Possible positive integer factor pairs of \(54\) are: \[ (1,54), (2,27), (3,18), (6,9) \] and their reverse pairs, since \(x\) and \(y\) are distinct. To determine \(x\), we need a unique value.

Step 1: Checking Statement (I).
Given: \[ x \text{ is odd} \] Possible odd values of \(x\): \[ x=1,\;3,\;27 \] with corresponding values: \[ (1,54), (3,18), (27,2) \] Multiple possibilities exist. So, Statement (I) alone is not sufficient.

Step 2: Checking Statement (II).
Given: \[ x>y \] Possible factor pairs where \(x>y\): \[ (54,1), (27,2), (18,3), (9,6) \] Again, multiple possibilities exist. So, Statement (II) alone is not sufficient.

Step 3: Checking both statements together.
From Statement (I), \(x\) must be odd. From Statement (II), \(x>y\). Possible pairs satisfying both: \[ (27,2) \] and \[ (9,6) \] Both satisfy: \[ xy=54 \] \(x\) is odd and \(x>y\). Thus, \(x\) is not uniquely determined. Hence, additional data are required.