If the six-digit number 479 xyz is exactly divisible by 7, 11 and 13, then `{( y + z) :x}` is equal to :

Options

A. `11/9`
B. `4`
C. `13/7`
D. `7/13`

Answer

To solve the problem, we need to determine the values of \( x \), \( y \), and \( z \) in the six-digit number \( 479xyz \) such that the number is divisible by 7, 11, and 13.

### Step-by-step Solution:

1. **Understand the Divisibility Condition**:
The number \( 479xyz \) must be divisible by \( 7 \), \( 11 \), and \( 13 \). The product of these three numbers is \( 1001 \). Therefore, \( 479xyz \) must be divisible by \( 1001 \).

2. **Determine the Range of the Number**:
The number \( 479xyz \) can be expressed as:
\[ 479000 + 100x + 10y + z \]
Since \( 479000 \) is already a six-digit number, we need to find the smallest and largest six-digit numbers divisible by \( 1001 \) that are greater than or equal to \( 479000 \).

3. **Find the Smallest Six-Digit Number Divisible by 1001**:
To find the smallest multiple of \( 1001 \) that is greater than or equal to \( 479000 \):
\[ \text{Smallest } n = \lceil \frac{479000}{1001} \rceil = 479 \]
Thus, the smallest six-digit number is:
\[ 1001 \times 479 = 479479 \]

4. **Find the Largest Six-Digit Number Divisible by 1001**:
To find the largest multiple of \( 1001 \) that is less than \( 480000 \):
\[ \text{Largest } n = \lfloor \frac{480000}{1001} \rfloor = 479 \]
Thus, the largest six-digit number is:
\[ 1001 \times 480 = 480480 \]

5. **Identify the Values of x, y, z**:
The valid six-digit numbers in the range are \( 479479 \) and \( 479480 \). We can see that:
- For \( 479479 \), \( x = 4 \), \( y = 7 \), \( z = 9 \).
- For \( 479480 \), \( x = 4 \), \( y = 8 \), \( z = 0 \).

6. **Calculate \( (y + z) : x \)**:
Using the values from \( 479479 \):
\[ y + z = 7 + 9 = 16 \]
\[ x = 4 \]
Thus, the ratio is:
\[ \frac{y + z}{x} = \frac{16}{4} = 4 \]

### Final Answer:
The value of \( (y + z) : x \) is \( 4 \).