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?

Statement I alone: 2x + y is an integer. This doesn't tell whether x + y is integer (e.g., x = 0.5, y = 0.5 gives 2x+y = 1.5 not integer; try x = 0.5, y = 0 gives 2x+y = 1 integer but x+y = 0.5 not integer). So Statement I alone insufficient. Statement II alone: x + 2y is an integer. Similarly insufficient (x = 0, y = 0.5 gives x+2y = 1 integer, x+y = 0.5 not integer). Combining: adding the two integers gives (2x+y) + (x+2y) = 3x + 3y = 3(x+y), which is an integer. So 3(x+y) is an integer, meaning x+y could be of the form k/3 (rational with denominator dividing 3). For example x = 1/3, y = 1/3 gives 2x+y = 1 integer ✓, x+2y = 1 integer ✓, but x+y = 2/3 not integer. So even both together do not guarantee x+y is integer. Hence the answer is (d), the question cannot be answered even using both statements together.