Consider the Question and two Statements given below : Question : Is x an integer? Statement-1 : x/3 is not an integer. Statement-2 : 3x is an integer. Which one of the following is correct in respect of the Question and the Statements?

(a) Statement-1 alone is sufficient to answer the Question
(b) Statement-2 alone is sufficient to answer the Question
(c) Both Statement-1 and Statement-2 are sufficient to answer the Question
(d) Both Statement-1 and Statement-2 are not sufficient to answer the Question ✓ UPSC's answer

Why the answer is (d)

• Statement-1 states that x/3 is not an integer, which implies x is not a multiple of 3, but x could still be an integer (e.g., x=1) or a non-integer (e.g., x=1/2), so it is insufficient alone.

• Statement-2 states that 3x is an integer, which implies x is a rational number with a denominator that divides 3, but x could be an integer (e.g., x=1, so 3x=3) or a non-integer (e.g., x=1/3, so 3x=1), so it is insufficient alone.

• Combining both statements, we know x is not a multiple of 3 and 3x is an integer.

• Consider x=1: x/3 = 1/3 (not an integer) and 3x = 3 (an integer). Here, x is an integer.

• Consider x=1/3: x/3 = 1/9 (not an integer) and 3x = 1 (an integer). Here, x is not an integer.

• Since both an integer and a non-integer satisfy both statements, the question cannot be answered definitively, making both statements together insufficient.

Why the other options are wrong

(a) Statement-1 alone is sufficient to answer the Question: Statement-1 alone allows x to be either an integer (like 1) or a non-integer (like 1/2), so it does not provide a unique answer.
(b) Statement-2 alone is sufficient to answer the Question: Statement-2 alone allows x to be either an integer (like 1) or a non-integer (like 1/3), so it does not provide a unique answer.
(c) Both Statement-1 and Statement-2 are sufficient to answer the Question: Even when combined, the statements are satisfied by both x=1 (an integer) and x=1/3 (a non-integer), so they are not sufficient to determine if x is an integer.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.