If `m + n = mn - 5`, then the maximum number of ordered pairs of (m,n) for every m, n is a natural number.

Options

A. 4
B. 5
C. 6
D. none of these

Answer

To solve the equation \( m + n = mn - 5 \) for natural numbers \( m \) and \( n \), we can rearrange the equation and analyze it step by step.

### Step 1: Rearranging the Equation
We start with the equation:
\[ m + n = mn - 5 \]
Rearranging gives us:
\[ mn - m - n = 5 \]
Adding 1 to both sides:
\[ mn - m - n + 1 = 6 \]
This can be factored as:
\[ (m - 1)(n - 1) = 6 \]

### Step 2: Finding Factor Pairs of 6
Next, we need to find the factor pairs of 6 since \( (m - 1)(n - 1) = 6 \). The positive factor pairs of 6 are:
1. \( (1, 6) \)
2. \( (2, 3) \)
3. \( (3, 2) \)
4. \( (6, 1) \)

### Step 3: Calculating \( m \) and \( n \) from Factor Pairs
Now we will convert these factor pairs back to \( m \) and \( n \):
1. For the pair \( (1, 6) \):
- \( m - 1 = 1 \) and \( n - 1 = 6 \) gives \( m = 2 \) and \( n = 7 \).

2. For the pair \( (2, 3) \):
- \( m - 1 = 2 \) and \( n - 1 = 3 \) gives \( m = 3 \) and \( n = 4 \).

3. For the pair \( (3, 2) \):
- \( m - 1 = 3 \) and \( n - 1 = 2 \) gives \( m = 4 \) and \( n = 3 \).

4. For the pair \( (6, 1) \):
- \( m - 1 = 6 \) and \( n - 1 = 1 \) gives \( m = 7 \) and \( n = 2 \).

### Step 4: Listing the Ordered Pairs
From the calculations, we have the following ordered pairs:
1. \( (2, 7) \)
2. \( (3, 4) \)
3. \( (4, 3) \)
4. \( (7, 2) \)

### Conclusion
Thus, the maximum number of ordered pairs \( (m, n) \) where both \( m \) and \( n \) are natural numbers is **4**.