UPSC CSE 2024 · PYQ · Data Sufficiency / Number Properties · medium
A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: What are the unique values of x and y, where x, y are distinct natural numbers? Statement-I: x / y is odd. Statement-II: xy = 12. Which one of the following is correct in respect of the above Question and the Statements?
A.The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
B.The Question can be answered by using either Statement alone
C.The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone✓ Correct
D.The Question cannot be answered even by using both the Statements together
Explanation
Statement I alone (x/y is odd, with x, y distinct natural numbers) has infinitely many solutions like (3,1), (5,1), (9,3), etc., so not unique. Statement II alone gives factor pairs of 12: (1,12), (2,6), (3,4), (4,3), (6,2), (12,1) — multiple distinct pairs, so not unique. Combining both: xy = 12 and x/y is an odd integer. Checking pairs where x/y is an integer and odd: (12,1) gives 12 (even), (6,2) gives 3 (odd) ✓, (4,3) gives 4/3 (not integer), (3,4), (2,6), (1,12) give fractions. So only (x, y) = (6, 2) works, giving unique values. Thus both statements together are required. The answer is (c).
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