If r is the remainder when each of 6454, 7306 and 8797 is divided by the greatest number d(d > 1), then (d—r) is equal to:

Options

A. 126
B. 64
C. 137
D. 149

Answer

To solve the problem, we need to find the greatest number \( d \) such that when each of the numbers 6454, 7306, and 8797 is divided by \( d \), they all leave the same remainder \( r \).

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find a number \( d \) such that \( 6454 \mod d = r \), \( 7306 \mod d = r \), and \( 8797 \mod d = r \). This implies that the differences between these numbers must be divisible by \( d \).

2. **Calculate the Differences**:
- Difference between 7306 and 6454:
\[ 7306 - 6454 = 852 \]
- Difference between 8797 and 7306:
\[ 8797 - 7306 = 1491 \]
- Difference between 8797 and 6454:
\[ 8797 - 6454 = 2343 \]

3. **Find the GCD**: The greatest number \( d \) must divide all these differences. Thus, we need to find the GCD of 852, 1491, and 2343.

4. **Calculate GCD of 852 and 1491**:
- Prime factorization of 852:
\[ 852 = 2^2 \times 3 \times 71 \]
- Prime factorization of 1491:
\[ 1491 = 3 \times 497 = 3 \times 7 \times 71 \]
- GCD(852, 1491) = \( 3 \times 71 = 213 \)

5. **Calculate GCD of 213 and 2343**:
- Prime factorization of 2343:
\[ 2343 = 3 \times 781 = 3 \times 11 \times 71 \]
- GCD(213, 2343) = \( 3 \times 71 = 213 \)

6. **Determine \( r \)**: Now, we need to find the remainder \( r \) when any of the original numbers is divided by \( d \). Let's use \( d = 213 \):
- Calculate \( r \):
\[ r = 6454 \mod 213 \]
- Performing the division:
\[ 6454 \div 213 \approx 30.3 \quad \text{(take the integer part 30)} \]
\[ 30 \times 213 = 6390 \]
\[ r = 6454 - 6390 = 64 \]

7. **Calculate \( d - r \)**:
\[ d - r = 213 - 64 = 149 \]

### Final Answer:
Thus, \( d - r = 149 \).