If the 6-digit number 479xy9 is divisible by each of 7,11,13, then (x-y) is equal to -

Options

A. `-3`
B. 2
C. 3
D. `-2`

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) in the 6-digit number \( 479xy9 \) such that it is divisible by 7, 11, and 13.

### Step-by-step Solution:

1. **Finding the LCM of 7, 11, and 13**:
- The numbers 7, 11, and 13 are all prime numbers. Therefore, their least common multiple (LCM) is simply their product:
\[ \text{LCM} = 7 \times 11 \times 13 = 1001 \]

2. **Understanding the divisibility condition**:
- For the number \( 479xy9 \) to be divisible by 1001, it must be a multiple of 1001. We can express this condition as:
\[ 479xy9 \equiv 0 \mod 1001 \]

3. **Finding the range of \( 479xy9 \)**:
- The number \( 479xy9 \) can be expressed as \( 479000 + 10000x + 1000y + 9 \).
- We need to find suitable values for \( x \) and \( y \) such that \( 479xy9 \) is a multiple of 1001.

4. **Calculating the nearest multiples of 1001**:
- We can find the nearest multiples of 1001 around 479000. The closest multiples can be calculated as:
\[ \text{Multiple 1} = 1001 \times 479 = 479479 \]
\[ \text{Multiple 2} = 1001 \times 480 = 480480 \]

5. **Finding suitable \( x \) and \( y \)**:
- The number \( 479xy9 \) must be between 479479 and 480480. Thus, we can set \( 479xy9 = 479479 \) and check for \( x \) and \( y \):
- From \( 479479 \), we can see that \( x = 4 \) and \( y = 7 \).

6. **Calculating \( x - y \)**:
- Now that we have \( x = 4 \) and \( y = 7 \), we can calculate:
\[ x - y = 4 - 7 = -3 \]

### Final Answer:
Thus, \( x - y = -3 \).