To find the three-digit number $n$ that, when dividing $12284$ and $7656$, leaves the same remainder, we need to consider the following:

If $12284$ divided by $n$ leaves a remainder $r$, and $7656$ divided by $n$ leaves the same remainder $r$, then:

$12284 = kn + r$ $7656 = mn + r$

where $k$ and $m$ are the quotients when $12284$ and $7656$ are divided by $n$, respectively.

By subtracting the second equation from the first, we eliminate $r$:

$12284 - 7656 = kn + r - (mn + r)$

Simplifying this, we get:

$12284 - 7656 = (k - m)n$

$4628 = (k - m)n$

This means that $n$ must be a divisor of $4628$. Now, let's find the three-digit divisors of $4628$.It appears that there are no three-digit divisors of $4628$. This suggests that no three-digit number $n$ exists such that when dividing $12284$ and $7656$, the same remainder is left.

If you have any further questions or need additional details, please let me know!

Here are some related questions that might interest you:

1. What are the divisors of a given number?
2. How can we find the greatest common divisor (GCD) of two numbers?
3. How do we solve problems involving modular arithmetic?
4. What are the methods to solve problems involving remainders?
5. Can there be more complicated cases of finding common divisors in different contexts?

Tip: When dealing with problems involving divisors and remainders, always start by simplifying the problem and finding common patterns or relationships between the numbers involved.