# Q: 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

**Explanation : **

**Objective: Determine if (x + y) is an integer.**

Analysis:

1. Solving the System:

Let 2x + y = A (where A is an integer).

Let x + 2y = B (where B is an integer).

Multiply equation (1) by 2:

4x + 2y = 2A

Subtract equation (2):

3x = 2A – B

x = (2A – B)/3

Substitute x back:

y = A – 2x = A – 2(2A – B)/3 = (-A + 2B)/3

Therefore:

x + y = (2A – B)/3 + (-A + 2B)/3 = (A + B)/3

For (x + y) to be integer, A + B must be divisible by 3.

**2. Evaluating Statements:**

Using Statement I Alone: Knowing 2x + y is integer (A) doesn’t guarantee x + y is integer.

Using Statement II Alone: Knowing x + 2y is integer (B) doesn’t guarantee x + y is integer.

Using Both Statements Together: Even with both A and B known, x + y = (A + B)/3 is not guaranteed to be integer unless A + B is divisible by 3, which isn’t assured.

**Conclusion:**

Even with both statements, there’s insufficient information to determine whether (x + y) is an integer.

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