Consider a 3-digit number. Question : What is the number ? Statement-1 : The sum of the digits of the number is equal to the product of the digits. Statement-2 : The number is divisible by the sum of the digits of the number. Which one of the following is correct in respect of the above Question and the Statements ?
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
- (b) The Question can be answered by using either Statement alone.
- (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
- (d) The Question cannot be answered even by using both the Statements together. ✓ UPSC's answer
Why the answer is (d)
• Let the digits of the 3-digit number be $a, b, c$ where $a \in \{1, \dots, 9\}$ and $b, c \in \{0, \dots, 9\}$.
• Statement 1 states $a + b + c = a \times b \times c$. If any digit is 0, the product is 0, implying the sum is 0, which is impossible for a 3-digit number. Thus, all digits are non-zero.
• Testing small integers for Statement 1: If digits are 1, 2, 3, sum = 6 and product = 6. This satisfies the condition. The possible numbers are permutations of 1, 2, 3: 123, 132, 213, 231, 312, 321.
• Statement 2 states the number is divisible by the sum of its digits. For the set {1, 2, 3}, the sum is 6. All permutations (123, 132, 213, 231, 312, 321) are divisible by 6 (since they are even and sum of digits is divisible by 3).
• Since multiple numbers (e.g., 123 and 132) satisfy both statements simultaneously, the specific number cannot be uniquely determined.
Why the other options are wrong
- (a) The Question can be answered by using one of the Statements alone, but cannot be answered…
- Neither statement alone provides a unique number; Statement 1 yields six candidates and Statement 2 is satisfied by all of them.
- (b) The Question can be answered by using either Statement alone.
- Statement 1 alone yields six possible numbers (permutations of 1, 2, 3), so it cannot answer the question uniquely.
- (c) The Question can be answered by using both the Statements together, but cannot be…
- Using both statements together still leaves six possible numbers (123, 132, 213, 231, 312, 321), so the question cannot be answered uniquely.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.