Directions : Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. What is the value of the two-digit number? I. The product of the digits is 72 and the difference between the digits is 1. II. The digit at the unit place is greater than the other.

## Concept & Logic To uniquely identify a two-digit number, we must find the exact mathematical values of both digits and determine their correct order (which digit is in the tens place and which is in the units place). ## Step-by-Step Solution * **Analyze Statement I:** The product of the digits is 72, and their difference is 1. The only two single-digit integers that multiply to 72 and have a difference of 1 are 8 and 9 ($8 \times 9 = 72$, $9 - 8 = 1$). This means the two-digit number can be either 89 or 98. Because there are two possible answers, Statement I alone is not sufficient. * **Analyze Statement II:** The digit at the unit place is greater than the other. This gives a condition for the order of the digits, but provides no numerical values. There are many numbers that fit this description (e.g., 12, 13, 29, 34). Statement II alone is not sufficient. * **Combine Statements I & II:** From Statement I, the possible numbers are restricted to 89 or 98. From Statement II, the unit digit must be strictly greater than the tens digit. In the number 89, the unit digit (9) is greater than the tens digit (8). This satisfies the condition. In the number 98, the unit digit (8) is less than the tens digit (9). This violates the condition. Applying Statement II eliminates 98, leaving exactly one unique answer: 89. ## Exam Strategy & Shortcut Quickly factor the product 72 into single digits mentally: 8 and 9. Notice that Statement I gives two permutations (89, 98). Look at Statement II; it is a strict inequality regarding digit positions. This acts as a perfect tie-breaker. You can immediately conclude that combining both statements will yield a unique answer without writing out full equations. ## Common Pitfall A frequent mistake is solving Statement I and prematurely assuming it is sufficient, ignoring that "89" and "98" represent two entirely distinct values in standard arithmetic. ## Final Answer **Therefore, the correct answer is Data in both Statements I and II together are necessary.**