Q. 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.

d) The Question cannot be answered even by using both the Statements together.

Correct answer: c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

To determine the unique values of x and y (distinct natural numbers) that satisfy the given conditions, let’s analyze each statement individually and then consider them together.

## Statement I: x/y is odd.

Interpretation: For x/y to be an odd integer, x must be an odd multiple of y. This implies that y must be a divisor of x, and the quotient x/y must be an odd number.

Limitations: Without additional information, there are multiple pairs (x, y) that can satisfy this condition. For example:

– x = 3, y = 1 (since 3/1 = 3 is odd)

– x = 9, y = 3 (since 9/3 = 3 is odd)

Conclusion: Statement I alone is insufficient to determine unique values of x and y.

### Statement II: xy = 12.

Interpretation: The product of x and y is 12. Considering that x and y are distinct natural numbers, the possible pairs are:

– (1, 12)

– (2, 6)

– (3, 4)

Limitations: Multiple valid pairs satisfy this condition.

Conclusion: Statement II alone is insufficient to determine unique values of x and y.

Combining Statements I and II:

Now, let’s use both statements together.

1. From Statement II, the possible pairs are (1, 12), (2, 6), and (3, 4).

2. Applying Statement I (x/y is odd) to each pair:

– (1, 12): 1/12 is not an integer, so this pair is invalid.

– (2, 6): 2/6 = 1/3 is not an integer, so this pair is invalid.

– (3, 4): 3/4 is not an integer, so this pair appears invalid at first glance.

However, there’s a misunderstanding here. For x/y to be an odd integer, x must be a multiple of y, and the quotient must itself be odd.

Re-examining the possible pairs:

– (12, 1): 12/1 = 12 (Even)

– (6, 2): 6/2 = 3 (Odd)

– (4, 3): 4/3 is not an integer.

Valid Pair: (6, 2) since 6/2 = 3 is an odd integer.

Conclusion: Using both statements together, the unique values are x = 6 and y = 2.

Therefore, option c is the correct choice.