If r is the remainder when each of 4749, 5601 and 7092 is divided by the greatest possible number `d(gt1)`, then the value of (d + r) will be:

Options

A. 276
B. 271
C. 298
D. 282

Answer

To solve the problem, we need to find the greatest possible number \( d \) such that when each of the numbers 4749, 5601, and 7092 is divided by \( d \), they leave the same remainder \( r \). Then, we will compute \( d + r \).

### Step-by-step Solution:

1. **Identify the Numbers**: We have three numbers:
- \( a = 4749 \)
- \( b = 5601 \)
- \( c = 7092 \)

2. **Calculate the Differences**: To find the greatest common divisor (GCD), we first calculate the differences between the numbers:
- \( b - a = 5601 - 4749 = 852 \)
- \( c - b = 7092 - 5601 = 1491 \)
- \( c - a = 7092 - 4749 = 2343 \)

3. **List the Differences**: The differences we have are:
- \( 852 \)
- \( 1491 \)
- \( 2343 \)

4. **Find the GCD of the Differences**: We need to find the GCD of these three differences:
- First, find \( \text{GCD}(852, 1491) \):
- The prime factorization of \( 852 = 2^2 \times 3 \times 71 \)
- The prime factorization of \( 1491 = 3 \times 497 \) (where \( 497 = 7 \times 71 \))
- Thus, \( \text{GCD}(852, 1491) = 3 \times 71 = 213 \)

- Next, find \( \text{GCD}(213, 2343) \):
- The prime factorization of \( 2343 = 3 \times 781 \) (where \( 781 = 11 \times 71 \))
- Thus, \( \text{GCD}(213, 2343) = 3 \times 71 = 213 \)

Therefore, the greatest possible number \( d \) is \( 213 \).

5. **Calculate the Remainder \( r \)**: Now we need to find the remainder when each number is divided by \( d \):
- For \( a = 4749 \):
\[ 4749 \div 213 = 22 \quad \text{(whole part)} \quad \Rightarrow \quad 22 \times 213 = 4686 \]
\[ r = 4749 - 4686 = 63 \]

- For \( b = 5601 \):
\[ 5601 \div 213 = 26 \quad \Rightarrow \quad 26 \times 213 = 5538 \]
\[ r = 5601 - 5538 = 63 \]

- For \( c = 7092 \):
\[ 7092 \div 213 = 33 \quad \Rightarrow \quad 33 \times 213 = 6999 \]
\[ r = 7092 - 6999 = 93 \quad \text{(this is incorrect, we need to check the GCD again)} \]

After checking, we find that the remainders are consistent and \( r = 63 \).

6. **Calculate \( d + r \)**:
\[ d + r = 213 + 63 = 276 \]

### Final Answer:
The value of \( d + r \) is \( 276 \).