Two numbers 11284 and 7655, when divided by a certain number of three digits, leaves the same remainder. The sum of digits of such a three-digit number is

Options

A. a) 8
B. b) 9
C. c) 10
D. d) 11

Answer

To solve the problem step-by-step, we need to find a three-digit number that leaves the same remainder when dividing the two numbers, 11284 and 7655. Here’s how we can approach the solution:

### Step 1: Find the Difference
First, we calculate the difference between the two numbers:
\[ 11284 - 7655 = 3629 \]

### Step 2: Find the Factors of the Difference
Next, we need to find the factors of 3629. A number that leaves the same remainder when dividing both 11284 and 7655 must be a factor of their difference.

### Step 3: Check for Factors
We will check for factors of 3629. We start by checking divisibility by smaller prime numbers:
- **1**: 3629 is divisible by 1.
- **2**: 3629 is odd, so it is not divisible by 2.
- **3**: The sum of the digits (3 + 6 + 2 + 9 = 20) is not divisible by 3, so 3629 is not divisible by 3.
- **4**: The last two digits (29) do not form a number divisible by 4, so it is not divisible by 4.
- **5**: 3629 does not end in 0 or 5, so it is not divisible by 5.
- **7**: \(3629 \div 7 \approx 518.43\) (not a whole number).
- **11**: \(3629 \div 11 \approx 329.91\) (not a whole number).
- **19**: \(3629 \div 19 = 191\) (this is a whole number).

### Step 4: Identify the Three-Digit Factor
From our calculations, we find that 191 is a factor of 3629. Since 191 is a three-digit number, it is a candidate for our solution.

### Step 5: Calculate the Sum of the Digits
Now, we need to find the sum of the digits of the number 191:
\[ 1 + 9 + 1 = 11 \]

### Final Answer
Thus, the sum of the digits of the three-digit number is:
\[ \text{Sum of digits} = 11 \]