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 the Statements together, but cannot be answered using either Statement alone
(d) The Question cannot be answered even by using both the Statements together ✓ UPSC's answer

Why the answer is (d)

• Let 2x + y = m and x + 2y = n. Adding: 3(x + y) = m + n, so x + y = (m + n)/3.

• Even when m and n are integers, (m + n)/3 need not be one.

• Example: x = y = 1/3 gives 2x + y = 1 and x + 2y = 1, but x + y = 2/3. With x = y = 1, all three are integers.

• So the question cannot be answered even with both statements — option (d).

Why the other options are wrong

(a) The Question can be answered by using one of the Statements alone, but cannot be answered…: Neither statement alone provides sufficient information to determine if x + y is an integer.
(b) The Question can be answered by using either Statement alone: Neither statement alone is sufficient to answer the question definitively.
(c) The Question can be answered by using both the Statements together, but cannot be…: Even using both statements together, the result depends on whether the sum of the two integers is divisible by 3, which is not guaranteed.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.