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 ✓ UPSC's answer
(d) The Question cannot be answered even by using both the Statements together

Why the answer is (c)

• Statement I states that x/y is odd, which implies x and y are both odd or both even, but does not provide specific values or a finite set of pairs.

• Statement II states that xy = 12, which yields the distinct natural number pairs (1, 12), (2, 6), and (3, 4).

• Using Statement I alone, there are infinitely many pairs (e.g., 3/1, 5/1, 4/2), so the unique values cannot be determined.

• Using Statement II alone, there are three possible pairs, so the unique values cannot be determined.

• Combining both statements, we test the pairs from Statement II against Statement I: 1/12 is not an integer, 2/6 is not an integer, and 4/3 is not an integer; however, the condition 'x/y is odd' typically implies x/y is an odd integer in such contexts, or we check if any pair satisfies it. Wait, let's re-evaluate. If x/y is odd, x must be a multiple of y such that the quotient is odd. From xy=12, possible (x,y) are (12,1), (6,2), (4,3), (3,4), (2,6), (1,12). x/y is odd integer only if x=12, y=1 (12/1=12, even), x=6, y=2 (6/2=3, odd). So (6,2) works. Are there others? x=3, y=1 is not in xy=12. x=12, y=1 gives 12 (even). x=6, y=2 gives 3 (odd). x=4, y=1 not in xy=12. x=2, y=1 not in xy=12. So only (6,2) satisfies both. Thus, unique values are x=6, y=2.

• Therefore, the question can be answered only by using both statements together.

Why the other options are wrong

(a) The Question can be answered by using one of the Statements alone, but cannot be answered…: Statement I alone allows infinite solutions, and Statement II alone allows multiple solutions, so neither alone is sufficient.
(b) The Question can be answered by using either Statement alone: Neither statement alone provides a unique solution; Statement I has infinite pairs and Statement II has three pairs.
(d) The Question cannot be answered even by using both the Statements together: The combination of both statements yields a unique pair (x=6, y=2), so the question is answerable.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.