If a number 676xy is divisible by 3, 7, and 11 then find the value of (3x - 5y).

Options

A. 4
B. 7
C. 6
D. 9

Answer

To solve the problem, we need to determine the values of \(x\) and \(y\) such that the number \(676xy\) is divisible by \(3\), \(7\), and \(11\). The least common multiple of these numbers is \(231\).

### Step-by-Step Solution:

1. **Identify the number**: The number we are considering is \(676xy\).

2. **Determine divisibility by 231**: We need to check the divisibility of \(676xy\) by \(231\).

3. **Calculate \(67600\) to \(67699\)**: We can check the values of \(67600\) to \(67699\) for divisibility by \(231\).

4. **Perform the division**:
- First, calculate \(67600 \div 231\):
\[ 67600 \div 231 \approx 292.64 \quad \text{(take the integer part, which is 292)} \]
- Now, calculate \(292 \times 231\):
\[ 292 \times 231 = 67452 \]
- Find the remainder:
\[ 67600 - 67452 = 148 \]

5. **Check next integer**: Now check \(293\):
- Calculate \(293 \times 231\):
\[ 293 \times 231 = 67683 \]
- Find the remainder:
\[ 67600 - 67683 = -83 \quad \text{(not applicable since we need a positive remainder)} \]

6. **Check \(67699\)**:
- Calculate \(67699 \div 231\):
\[ 67699 \div 231 \approx 293.45 \quad \text{(take the integer part, which is 293)} \]
- Calculate \(293 \times 231\):
\[ 293 \times 231 = 67683 \]
- Find the remainder:
\[ 67699 - 67683 = 16 \]

7. **Determine \(xy\)**: Since \(676xy\) must be divisible by \(231\), we need \(xy\) such that:
\[ 676xy = 67600 + 10x + y \equiv 0 \mod 231 \]
- We need \(10x + y \equiv -16 \mod 231\).

8. **Finding \(x\) and \(y\)**:
- We can try different values for \(x\) and \(y\) to satisfy this condition.
- After testing combinations, we find that \(x = 8\) and \(y = 3\) satisfy the condition.

9. **Calculate \(3x - 5y\)**:
\[ 3x - 5y = 3(8) - 5(3) = 24 - 15 = 9 \]

### Final Answer:
The value of \(3x - 5y\) is \(9\).