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 3-digit number have digits a, b, c (a from 1 to 9, b and c from 0 to 9). Statement 1 requires a + b + c = a × b × c. If any digit is 0 the product is 0 while the sum is at least 1, so all digits are at least 1. Setting one digit to 1 and the other two to p and q gives 1 + p + q = pq, i.e. (p−1)(q−1) = 2, yielding {p, q} = {2, 3}. With no digit equal to 1, a systematic check of a, b, c ≥ 2 shows no solution. Hence the digits must be a permutation of {1, 2, 3}, giving exactly six candidates: 123, 132, 213, 231, 312, 321. Statement 1 alone leaves six possibilities, so it cannot determine a unique number.

• Statement 2 requires the number to be divisible by the sum of its digits. This condition is satisfied by a large set of 3-digit numbers (for instance 100, 102, 111, 112, 126, 132, 312, …), so Statement 2 alone also cannot pin down a single number.

• Combining both statements, every candidate from Statement 1 has digit-sum 6, so we test divisibility by 6: 123 ÷ 6 = 20.5 (no), 132 ÷ 6 = 22 (yes), 213 ÷ 6 = 35.5 (no), 231 ÷ 6 = 38.5 (no), 312 ÷ 6 = 52 (yes), 321 ÷ 6 = 53.5 (no).

• Two numbers, 132 and 312, satisfy both statements simultaneously, so even together the statements do not yield a unique answer.

• Because the question asks for "the number" (a single value) and two distinct values remain after applying both statements, the question cannot be answered even by using both statements together, so the answer is option (d).

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 gives a unique number: Statement 1 leaves six candidates and Statement 2 is satisfied by many 3-digit numbers, so the question cannot be answered by one statement alone.
(b) The Question can be answered by using either Statement alone.: Neither statement alone is sufficient; Statement 1 yields six permutations of {1,2,3} and Statement 2 is met by a wide range of 3-digit numbers, so the question cannot be answered by either statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be…: Even using both statements together, two numbers (132 and 312) satisfy both conditions, so the question still cannot be answered uniquely; the statements together are insufficient, not merely individually insufficient.

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.