Let m be a positive integer and let the lines ` 13x + 11y = 700 ` and ` y= mx - 1 ` intersect in a point whose coordinates are integer. Then m equals to :

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the problem, we need to find the value of \( m \) such that the lines \( 13x + 11y = 700 \) and \( y = mx - 1 \) intersect at integer coordinates.

### Step 1: Rewrite the equations
We start with the two equations:
1. \( 13x + 11y = 700 \)
2. \( y = mx - 1 \)

We can rearrange the first equation to express \( y \) in terms of \( x \):
\[ 11y = 700 - 13x \]
\[ y = \frac{700 - 13x}{11} \]

### Step 2: Substitute for \( y \)
Now, we substitute \( y \) from the second equation into the rearranged first equation:
\[ \frac{700 - 13x}{11} = mx - 1 \]

### Step 3: Clear the fraction
To eliminate the fraction, we multiply both sides by 11:
\[ 700 - 13x = 11(mx - 1) \]
\[ 700 - 13x = 11mx - 11 \]

### Step 4: Rearrange the equation
Now, we rearrange the equation to isolate terms involving \( x \):
\[ 700 + 11 = 11mx + 13x \]
\[ 711 = 11mx + 13x \]
\[ 711 = x(11m + 13) \]

### Step 5: Solve for \( x \)
Now we can solve for \( x \):
\[ x = \frac{711}{11m + 13} \]

### Step 6: Ensure \( x \) is an integer
For \( x \) to be an integer, \( 11m + 13 \) must be a divisor of 711. We need to find the divisors of 711.

### Step 7: Find the divisors of 711
The prime factorization of 711 is:
\[ 711 = 3 \times 237 = 3 \times 3 \times 79 = 3^2 \times 79 \]
The divisors of 711 are: \( 1, 3, 9, 79, 237, 711 \).

### Step 8: Set up the equation
We set up the equation:
\[ 11m + 13 = d \quad \text{(where \( d \) is a divisor of 711)} \]
Rearranging gives:
\[ 11m = d - 13 \]
\[ m = \frac{d - 13}{11} \]

### Step 9: Check each divisor
We will check which divisors yield a positive integer \( m \):

1. For \( d = 1 \):
\[ m = \frac{1 - 13}{11} = \frac{-12}{11} \quad \text{(not positive)} \]

2. For \( d = 3 \):
\[ m = \frac{3 - 13}{11} = \frac{-10}{11} \quad \text{(not positive)} \]

3. For \( d = 9 \):
\[ m = \frac{9 - 13}{11} = \frac{-4}{11} \quad \text{(not positive)} \]

4. For \( d = 79 \):
\[ m = \frac{79 - 13}{11} = \frac{66}{11} = 6 \quad \text{(positive integer)} \]

5. For \( d = 237 \):
\[ m = \frac{237 - 13}{11} = \frac{224}{11} \quad \text{(not an integer)} \]

6. For \( d = 711 \):
\[ m = \frac{711 - 13}{11} = \frac{698}{11} \quad \text{(not an integer)} \]

### Conclusion
The only positive integer value of \( m \) that satisfies the condition is:
\[ \boxed{6} \]