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?

Correct Option: Both Statement-1 and Statement-2 are not sufficient to answer the Question

Let us evaluate the sufficiency of the given statements to determine if x is an integer. Statement-1: x / 3 is not an integer. This tells us x is not a multiple of 3, but x could still be an integer (e.g., if x = 2, x/3 = 2/3 not an integer; if x = 1.5, x/3 = 0.5 not an integer). Thus, Statement-1 alone is not sufficient. Statement-2: 3x is an integer. This means x could be an integer (e.g., x = 1 => 3x = 3) or a fraction with denominator 3 (e.g., x = 1/3 => 3x = 1, where 1/3 is not an integer). Thus, Statement-2 alone is not sufficient. Combining Statement-1 and Statement-2: Statement-2 gives 3x = k (where k is an integer), so x = k/3. Statement-1 states x/3 is not an integer, meaning (k/3)/3 = k/9 is not an integer. Can k/3 be an integer? Let k = 2, then 3x = 2 => x = 2/3 (not an integer, and x/3 = 2/9 not an integer). Let k = 1, then 3x = 1 => x = 1/3 (not an integer, x/3 = 1/9 not an integer). Let k = 3, then 3x = 3 => x = 1 (which IS an integer, but then x/3 = 1/3 is not an integer). Wait! If x = 1, 3x = 3 (integer) and x/3 = 1/3 (not an integer). If x = 2/3, 3x = 2 (integer) and x/3 = 2/9 (not an integer). In both cases, both statements are satisfied! Thus, even combined, we cannot determine uniquely whether x is an integer or not. Therefore, both Statement-1 and Statement-2 are not sufficient.