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UPSC Prelims 2023CSATBasic NumeracyData Sufficiency - Number Properties

Q69. 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.

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.✓ Correct

Detailed Solution

✓ Correct Answer: Option D

We need to find a unique 3-digit number. Statement 1: Sum of digits = Product of digits. For example, if the digits are 1, 2, 3, then 1+2+3=6 and 1*2*3=6. Numbers like 123, 132, 213, 231, 312, 321 all satisfy this.

Since there are multiple possibilities, Statement 1 alone is not sufficient. Statement 2: The number is divisible by the sum of its digits. For example, 102 (sum=3, 102/3=34) and 108 (sum=9, 108/9=12) both satisfy this.

Since there are multiple possibilities, Statement 2 alone is not sufficient. Combining both statements: We look for numbers where the sum of digits equals the product of digits, AND the number is divisible by this sum.

Using the example digits 1, 2, 3 (sum=6, product=6): - 132: Sum=6, Product=6. 132 is divisible by 6 (132/6=22). This number satisfies both. - 312: Sum=6, Product=6. 312 is divisible by 6 (312/6=52).

This number also satisfies both. Since both 132 and 312 satisfy both statements, we cannot determine a unique 3-digit number. Therefore, even using both statements together, the question cannot be answered.

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