When 3820, 4672 und 6163 are divided by the greatest number x, the remainder in each is the same. What is the quotient when x divides 1035?

Options

A. 6
B. 8
C. 4
D. 3

Answer

To solve the problem, we need to find the greatest number \( x \) such that when 3820, 4672, and 6163 are divided by \( x \), they all leave the same remainder. Then, we will find the quotient when \( x \) divides 1035.

### Step-by-Step Solution:

1. **Calculate the Differences**:
We first find the differences between the given numbers:
- \( 4672 - 3820 = 852 \)
- \( 6163 - 4672 = 1491 \)
- \( 6163 - 3820 = 2343 \)

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

3. **Find the Greatest Common Divisor (GCD)**:
We need to find the GCD of the differences \( 852, 1491, \) and \( 2343 \). We can do this using prime factorization.

- **Factorization of 852**:
- \( 852 = 2^2 \times 3 \times 71 \)

- **Factorization of 1491**:
- \( 1491 = 3 \times 7 \times 71 \)

- **Factorization of 2343**:
- \( 2343 = 3^2 \times 71 \)

4. **Identify the Common Factors**:
The common prime factors from the factorizations are:
- \( 3 \)
- \( 71 \)

Therefore, the GCD is:
\[ GCD = 3 \times 71 = 213 \]

5. **Determine \( x \)**:
Hence, the greatest number \( x \) is \( 213 \).

6. **Calculate the Quotient**:
Now, we need to find the quotient when \( 1035 \) is divided by \( 213 \):
\[ 1035 \div 213 = 4 \quad \text{(since } 213 \times 4 = 852\text{)} \]
The remainder is:
\[ 1035 - 852 = 183 \]

### Final Answer:
The quotient when \( x \) divides \( 1035 \) is \( 4 \).

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