Q64. A question is given followed by two Statements I and
II. Consider the Questions 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?

We need to find unique distinct natural numbers x and y. **Statement I: x/y is odd.** Possible pairs (x,y) where x/y is an odd natural number: - If y=1, x can be 3, 5, 7, ... (e.g., (3,1), (5,1)) - If y=2, x can be 6, 10, 14, ...

(e.g., (6,2), (10,2)) Since there are multiple possibilities, Statement I alone is insufficient. **Statement II: xy = 12.** Possible pairs (x,y) of distinct natural numbers: - (1,12), (2,6), (3,4), (4,3), (6,2), (12,1).

Since there are multiple possibilities, Statement II alone is insufficient. **Combining Statement I and Statement II:** We need a pair (x,y) from the list in Statement II where x/y is an odd natural number.

- (1,12): x/y = 1/12 (not an integer, so not odd) - (2,6): x/y = 2/6 = 1/3 (not an integer) - (3,4): x/y = 3/4 (not an integer) - (4,3): x/y = 4/3 (not an integer) - (6,2): x/y = 6/2 = 3 (which is an odd natural number). This gives (x=6, y=2).

- (12,1): x/y = 12/1 = 12 (which is an even natural number, not odd). The only pair that satisfies both conditions is (6,2). Thus, x=6 and y=2 are uniquely determined. Therefore, both statements together are sufficient. This question tests data sufficiency and number properties.