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. Digit in the ten's place is four times the digit in the unit's place. II. The two digits are not equal. III. Digit in the ten's place is cube of the digit in unit's place.

Concept & Logic We test each statement to find a unique two-digit number. The digits must be integers between 0 and 9, and the leading ten's digit cannot be 0. $$Original Number = 10x + y$$ Step-by-Step Solution * **Analyze Statement I:** The ten's digit ($x$) is $4 \times$ Unit's digit ($y$). The only valid single-digit integer pairs for $(x, y)$ are $(4,1)$ and $(8,2)$. The number could be 41 or 82. (Insufficient) * **Analyze Statement II:** The digits are unequal, $x \neq y$. (Insufficient) * **Analyze Statement III:** The ten's digit is the cube of the unit's digit, $x = y^3$. Since $x$ is a single non-zero digit, $y$ can only be 1 or 2. The possible pairs are $(1,1)$ or $(8,2)$. The number is 11 or 82. (Insufficient) * **Test Combo I & III:** Both statements are satisfied only by the unique number 82. (Sufficient) * **Test Combo II & III:** Statement III gives 11 or 82. Statement II says digits are unequal, thereby eliminating 11. The unique number is 82. (Sufficient) * Therefore, Statement III combined with either I or II is sufficient. * Since the option "III and either I or II" is not available among (a) to (d), we must choose "None of these". Exam Strategy & Shortcut Test discrete integer properties quickly. Cubes of digits 0-9 that result in single digits are extremely limited (just 0, 1, and 8). This severely restricts the possibilities, making checking statement combinations incredibly fast and visual. Common Pitfall A frequent mistake is forgetting that 11 is a valid two-digit number where the ten's digit ($1$) is indeed the cube of the unit's digit ($1^3 = 1$). Without remembering 11, you might falsely conclude that Statement III alone is sufficient to find 82. Final Answer **Therefore, the correct answer is None of these.**