Q68. A Question if 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?

We need to determine if (x+y) is an integer. **Statement I: (2x+y) is an integer.** - Let x = 0.5, y = 1. Then 2x+y = 2(0.5)+1 = 1+1 = 2 (integer). But x+y = 0.5+1 = 1.5 (not an integer).

- Let x = 1, y = 1. Then 2x+y = 2(1)+1 = 3 (integer). And x+y = 1+1 = 2 (integer). Since (x+y) can be an integer or not, Statement I alone is insufficient. **Statement II: (x+2y) is an integer.** - Let x = 1, y = 0.5.

Then x+2y = 1+2(0.5) = 1+1 = 2 (integer). But x+y = 1+0.5 = 1.5 (not an integer). - Let x = 1, y = 1. Then x+2y = 1+2(1) = 3 (integer). And x+y = 1+1 = 2 (integer). Since (x+y) can be an integer or not, Statement II alone is insufficient.

**Combining Statement I and Statement II:** We have: (2x+y) = K (an integer) and (x+2y) = M (an integer). Adding these two equations: (2x+y) + (x+2y) = K + M 3x + 3y = K + M 3(x+y) = K + M.

Since K and M are integers, K+M is also an integer. So, 3(x+y) is an integer. However, if 3(x+y) is an integer, (x+y) itself is not necessarily an integer. For example, if x=1/3 and y=1/3: - 2x+y = 2/3 + 1/3 = 1 (integer).

- x+2y = 1/3 + 2/3 = 1 (integer). - But x+y = 1/3 + 1/3 = 2/3 (not an integer). Therefore, even with both statements combined, we cannot definitively say if (x+y) is an integer.

This question tests data sufficiency and properties of integers and rational numbers.