Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the given question. What is the two-digit number? I. The number obtained by interchanging the digits of the number is greater than the original number by 18. II. Sum of the digits of the number is 14. III. Difference between the two digits of the number is 2.

Concept & Logic To identify a unique two-digit number, we need two distinct linear equations that provide a single, unambiguous solution for the ten's digit ($x$) and unit's digit ($y$). $$Original Number = 10x + y$$ Step-by-Step Solution * **Given Statement I:** The reversed number is greater by 18. * Calculation for I: $(10y + x) - (10x + y) = 18 \implies 9(y - x) = 18 \implies y - x = 2$. * **Given Statement II:** The sum of the digits is 14. So, $x + y = 14$. * **Given Statement III:** The difference between the digits is 2. So, $|x - y| = 2$. * **Test Combo I & II:** We have $y - x = 2$ and $x + y = 14$. Adding them yields $2y = 16 \implies y = 8$. Substituting back, $x = 6$. The unique number is 68. (Sufficient) * **Test Combo II & III:** We have $x + y = 14$ and $|x - y| = 2$. This yields two possible cases: $x-y=2$ (number is 86) and $y-x=2$ (number is 68). Since the answer is not a single unique number, this pair is insufficient. * **Test Combo I & III:** Both statements essentially describe the difference between the digits, lacking any information about their absolute sum or product. (Insufficient) Exam Strategy & Shortcut Pay close attention to "difference between digits" (Statement III) versus "reversed number difference" (Statement I). The reversed number difference always gives you the specific sign and order of the digits (e.g., $y > x$), whereas an absolute difference leads to multiple scenarios. Thus, combining a sum with a directional difference (I & II) will always be superior to combining a sum with an absolute difference (II & III). Common Pitfall A major trap is assuming Statements II and III are sufficient because they allow you to find the exact digits. While they give you the digits 6 and 8, they create two valid numbers: 68 and 86. Data Sufficiency rules dictate that a statement combination is only sufficient if it yields exactly *one* unique numerical answer. Final Answer **Therefore, the correct answer is Only I and II.**