Numbers 12284 and 8656 when divided by a certain 3 digit number leaves the same remainder .Find the three digit number

To solve this problem, we are looking for a three-digit number \( n \) that, when dividing both 12284 and 8656, leaves the same remainder.

### Step 1: Define the Problem Mathematically
Let the three-digit number be \( n \). We have:
- When 12284 is divided by \( n \), it leaves a remainder \( r \).
- When 8656 is divided by \( n \), it also leaves the same remainder \( r \).

We can write these as:
\[
12284 = q_1 \cdot n + r
\]
\[
8656 = q_2 \cdot n + r
\]
where \( q_1 \) and \( q_2 \) are the quotients.

### Step 2: Subtract the Two Equations
Subtracting the second equation from the first:
\[
12284 - 8656 = (q_1 \cdot n + r) - (q_2 \cdot n + r)
\]
Simplifying this, we get:
\[
12284 - 8656 = (q_1 - q_2) \cdot n
\]
\[
3628 = (q_1 - q_2) \cdot n
\]

So, \( n \) must be a divisor of 3628.

### Step 3: Find the Three-Digit Divisors of 3628
Let's find the divisors of 3628 that are three digits long.

The prime factorization of 3628 is:
\[
3628 = 2^2 \times 907
\]

The three-digit divisors are 907. 

### Step 4: Verify the Solution
We need to ensure 907 leaves the same remainder when dividing both numbers:

For 12284:
\[
12284 \div 907 = 13 \text{ remainder } 505
\]

For 8656:
\[
8656 \div 907 = 9 \text{ remainder } 505
\]

Since the remainders are the same, \( n = 907 \) is the correct three-digit number.

### Answer:
The three-digit number is **907**.

Would you like to see further details or have any questions about this solution?

Here are 5 related questions to explore further:
1. How can we find the divisors of any number systematically?
2. What is the significance of prime factorization in solving remainder problems?
3. How do we verify solutions in division and remainder problems?
4. Can we generalize this approach to find divisors for any set of numbers with equal remainders?
5. What other applications exist for finding numbers that leave the same remainder?

**Tip:** For problems involving remainders, subtracting the equations can often simplify the problem by removing the remainder and highlighting common divisors.

Numbers 12284 and 8656 when divided by a certain 3-digit number leave the same remainder. Find the three-digit number.

Solve the problem of finding the 3-digit divisor of 12284 and 8656 that leaves the same remainder. Learn about number theory, prime factorization, and divisibility in this step-by-step guide. Suitable for Grades 10-12.

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