the numbers 11284 and 7655 when divided by a certain number of three digits, leave the same remainder. Find the number and the remainder-

Options

A. 119,15
B. 191,15
C. 192, 52
D. 191, 51

Answer

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

### Step 1: Understand the problem
We need to find a three-digit number \( N \) such that:
- \( 11284 \mod N = r \)
- \( 7655 \mod N = r \)

This implies that the difference between the two numbers, \( 11284 - 7655 \), must be divisible by \( N \).

### Step 2: Calculate the difference
Calculate the difference between the two numbers:
\[ 11284 - 7655 = 3629 \]

### Step 3: Find the factors of the difference
Next, we need to find the factors of 3629 to identify possible values for \( N \). We can start by checking if 3629 is a prime number or if it can be factored.

### Step 4: Factor 3629
To factor 3629, we can check divisibility by prime numbers:
- 3629 is not divisible by 2 (it’s odd).
- Check divisibility by 3: \( 3 + 6 + 2 + 9 = 20 \) (not divisible).
- Check divisibility by 5: last digit is not 0 or 5 (not divisible).
- Check divisibility by 7, 11, 13, etc.

After testing, we find:
\[ 3629 = 19 \times 191 \]
This means the factors of 3629 are 19 and 191.

### Step 5: Identify the three-digit number
Since we are looking for a three-digit number, we take \( N = 191 \).

### Step 6: Calculate the remainder
Now we need to find the remainder when dividing both numbers by 191:
1. Calculate \( 11284 \mod 191 \):
\[ 11284 \div 191 \approx 59.1 \quad \Rightarrow \quad 191 \times 59 = 11269 \]
\[ 11284 - 11269 = 15 \]

2. Calculate \( 7655 \mod 191 \):
\[ 7655 \div 191 \approx 40.1 \quad \Rightarrow \quad 191 \times 40 = 7640 \]
\[ 7655 - 7640 = 15 \]

Both calculations give us a remainder of 15.

### Final Answer
Thus, the three-digit number is \( 191 \) and the common remainder is \( 15 \).

### Summary
- The three-digit number is \( 191 \).
- The remainder when both numbers are divided by \( 191 \) is \( 15 \).