# Q. 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.

Correct Answer: (d) The Question cannot be answered even by using both the Statements together.

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

Let’s take the 3-digit number as XYZ, where X, Y, and Z are its digits.

**Statement-1**:

## The sum of the digits of the number is equal to the product of the digits.

This means X+Y+Z = XYZ. There are multiple 3-digit numbers that satisfy this condition, such as 123, 132, 213, 231, 312, 321, etc.

**Statement-2**:

### The number is divisible by the sum of the digits of the number.

This means XYZ is divisible by (X+Y+Z). Again, there are multiple 3-digit numbers that satisfy this condition, such as 111, 222, 333, etc.

Even if we use both statements together, we still have multiple options for the 3-digit number. For example, both 111 and 222 satisfy both conditions. Therefore, even by using both statements together, we cannot definitively answer the question.