Find a three digit number which, when divided by 11284 and 7655 leaves the same remainder in each case.

Options

A. 199
B. 197
C. 193
D. 191

Answer

To find a three-digit number that, when divided by 11284 and 7655, leaves the same remainder in each case, we can follow these steps:

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

This means that \( N \) leaves the same remainder \( r \) when divided by both 11284 and 7655.

### Step 2: Set up the equations
Since \( N \) leaves the same remainder when divided by both numbers, we can express this as:
\[ N = 11284k + r \]
\[ N = 7655m + r \]
for some integers \( k \) and \( m \).

### Step 3: Eliminate \( r \)
By setting the two equations equal to each other, we can eliminate \( r \):
\[ 11284k + r = 7655m + r \]
This simplifies to:
\[ 11284k = 7655m \]
or
\[ 11284k - 7655m = 0 \]

### Step 4: Find the difference
To find a common multiple, we can also look at the difference between the two numbers:
\[ 11284 - 7655 = 3629 \]

### Step 5: Find the GCD
To find the three-digit number that satisfies the condition, we need to find the GCD of 11284 and 7655. We can use the Euclidean algorithm:
1. \( 11284 = 7655 \times 1 + 3629 \)
2. \( 7655 = 3629 \times 2 + 397 \)
3. \( 3629 = 397 \times 9 + 16 \)
4. \( 397 = 16 \times 24 + 13 \)
5. \( 16 = 13 \times 1 + 3 \)
6. \( 13 = 3 \times 4 + 1 \)
7. \( 3 = 1 \times 3 + 0 \)

The GCD is 1.

### Step 6: Find the required number
Since the GCD is 1, we can find a number that is a multiple of the difference \( 3629 \) and still fits within three digits. The multiples of 3629 are:
- \( 3629 \times 1 = 3629 \) (not a three-digit number)
- \( 3629 \times 0 = 0 \) (not a three-digit number)

Since the GCD is 1, we can also consider the smallest prime number that is less than 1000. The largest three-digit number that is less than 1000 is 997, which is prime.

### Step 7: Verify the result
We need to check if 997 leaves the same remainder when divided by 11284 and 7655:
- \( 997 \mod 11284 = 997 \)
- \( 997 \mod 7655 = 997 \)

Thus, both divisions leave the same remainder.

### Conclusion
The required three-digit number is **997**.

---