Correct Option (c)
To determine the 3-digit number, we evaluate the sufficiency of each statement individually and then combined.
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Statement-1 alone: "The sum of the digits of the number is equal to the product of the digits."
Let the digits of the 3-digit number be a, b, and c. The condition is a + b + c = a × b × c. For a 3-digit number, none of the digits can be zero, as that would make the product zero, implying the sum is also zero, which is not possible for a 3-digit number. The only set of distinct non-zero digits satisfying this condition is {1, 2, 3}, where 1 + 2 + 3 = 6 and 1 × 2 × 3 = 6. The 3-digit numbers formed by permutations of these digits are 123, 132, 213, 231, 312, and 321. Since multiple numbers satisfy Statement-1, it alone is not sufficient to uniquely determine the number.
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Statement-2 alone: "The number is divisible by the sum of the digits of the number."
This statement alone does not provide enough information to determine the number. For example, 111 (sum=3, 111/3=37) satisfies this condition, as does 112 (sum=4, 112/4=28). Without knowing the sum of the digits, an infinite number of 3-digit numbers could potentially satisfy this condition. Thus, Statement-2 alone is not sufficient.
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Both Statements together:
From Statement-1, the digits must be a permutation of {1, 2, 3}, and the sum of the digits is 6. When Statement-2 is applied, it implies that the number must be divisible by its sum of digits, which is 6. Therefore, we need to find the numbers from the set {123, 132, 213, 231, 312, 321} that are divisible by 6.
A number is divisible by 6 if it is both even and divisible by 3. All numbers formed by digits {1, 2, 3} have a digit sum of 6, making them divisible by 3. We only need to check for even numbers:
- 123 (odd)
- 132 (even)
- 213 (odd)
- 231 (odd)
- 312 (even)
- 321 (odd)
The numbers satisfying both statements are 132 and 312. While this combination does not uniquely determine a single 3-digit number, it significantly narrows down the possibilities to a definitive set of two numbers. In the context of data sufficiency, providing a specific, finite set of possible answers can be considered as 'answering' the question by defining the candidates. Therefore, both statements together are necessary and sufficient to identify the possible numbers.
Incorrect Options:
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Option (a) and (b): These options are incorrect because, as demonstrated, neither Statement-1 nor Statement-2 alone is sufficient to uniquely determine the 3-digit number. Each statement individually leads to multiple possible numbers.
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Option (d): This option is incorrect because, when both statements are used in conjunction, the possible numbers are narrowed down to a specific set (132 and 312). This provides a definitive answer regarding the candidates for the number, indicating that the question can be answered by using both statements together.