To solve this, let the three-digit number be denoted by $N$. According to the problem, when 12284 and 8655 are divided by $N$, they leave the same remainder. This implies that the difference between the two numbers must be divisible by $N$.

Step 1: Find the difference between the two numbers

$12284 - 8655 = 3629$

Thus, $N$ must be a divisor of 3629.

Step 2: Find the divisors of 3629

We now need to check whether 3629 has any divisors that are three-digit numbers. First, let's check if 3629 is a prime number.

• 3629 is not divisible by 2, 3, or 5 (checked by applying divisibility rules).
• Performing further checks, it turns out 3629 is not a prime number; it's divisible by 19.

Step 3: Factorization of 3629

$3629 \div 19 = 191$

So, $3629 = 19 \times 191$.

Step 4: Identify the three-digit divisor

Since 191 is the only factor of 3629 that is a three-digit number, the required value of $N$ is 191.

Thus, the three-digit number is $\boxed{191}$.

Would you like to see a detailed explanation of the steps?

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Here are 5 related questions:

1. How do you determine divisibility rules for small prime numbers?
2. Can you explain how to find the greatest common divisor (GCD) of two numbers?
3. What is the method to check if a number is prime?
4. How do you perform long division for large numbers?
5. How can modular arithmetic help in solving remainder problems?

Tip: When trying to find a common divisor between two numbers, always start by subtracting the smaller number from the larger one and factorizing the difference.