Data sufficiency - identifying a 3-digit number
Question
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?
Options
The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
The Question can be answered by using either Statement alone.
The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
The Question cannot be answered even by using both the Statements together.
Explanation
Evaluate the mathematical boundaries of the statements for a 3-digit number XYZ :
Statement 1: X + Y + Z = X × Y × Z. The only single-digit combinations that satisfy this unique property are permutations of the set \{1, 2, 3\} (since 1+2+3 = 1 × 2 × 3 = 6). This yields 6 possible 3-digit numbers: 123, 132, 213, 231, 312, 321. Insufficient.
Statement 2: The number is a multiple of its digit sum. Dozens of 3-digit numbers fit this rule (e.g., 100, 112, 120). Insufficient.
Combine both statements: Test which of our 6 candidates from Statement 1 are divisible by their digit sum (6) [cite: 4051, 4052]:
Since multiple distinct numbers satisfy both conditions, a unique solution cannot be isolated even when combined.
Answer: (d).
Question details
Year
2023
Paper
CSAT
Question
Q59
Section
Data Interpretation & Sufficiency
Sub-topic
Data Sufficiency
Type
Data sufficiency
Difficulty
Medium
Source hint
Data sufficiency
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