Q. A Question is given followed by two Statements I and II. Consider the Question and the Statements.

# Question: Is (x + y) an integer?

Statement-I: (2x+y) is an integer.

Statement-II: (x+2y) is an integer.

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

##### Question from UPSC Prelims 2024 CSAT Paper

**Explanation:**

**From Statement I:**

– If (2x + y) is an integer, it doesn’t necessarily mean that (x + y) is an integer.

– For example, x could be 0.5 and y could be 0, making (2x + y) = 1 (an integer), but (x + y) = 0.5 (not an integer).

**From Statement II:**

– Similarly, if (x + 2y) is an integer, it doesn’t necessarily mean that (x + y) is an integer.

– For example, x could be 0 and y could be 0.5, making (x + 2y) = 1 (an integer), but (x + y) = 0.5 (not an integer).

## Even if we combine both statements, we still can’t determine if (x + y) is an integer:

– x could be 1/3 and y could be 1/3. In this case:

(2x + y) = 2(1/3) + 1/3 = 1 (an integer)

(x + 2y) = 1/3 + 2(1/3) = 1 (an integer)

But (x + y) = 1/3 + 1/3 = 2/3 (not an integer)

Therefore, even with both statements, we cannot definitively answer whether (x + y) is an integer or not. The correct answer is indeed d) The Question cannot be answered even by using both the Statements together.