If m,n `in` N and `m=(n^2-n-35)/(n-4)`, then find the value of m.

Answer

To solve the equation \( m = \frac{n^2 - n - 35}{n - 4} \) where \( m, n \in \mathbb{N} \), we will follow these steps:

### Step 1: Factor the numerator
First, we need to factor the quadratic expression in the numerator \( n^2 - n - 35 \). We can look for two numbers that multiply to \(-35\) and add to \(-1\). These numbers are \( -7 \) and \( 5 \).

So, we can factor the numerator as:
\[ n^2 - n - 35 = (n - 7)(n + 5) \]

### Step 2: Rewrite the expression for \( m \)
Now, substituting the factored form into the expression for \( m \):
\[ m = \frac{(n - 7)(n + 5)}{n - 4} \]

### Step 3: Simplify the expression
Next, we can simplify the expression. However, we need to check if \( n - 4 \) is a factor of the numerator. Since \( n - 4 \) is not a factor of \( (n - 7)(n + 5) \), we will leave it as is.

### Step 4: Analyze the expression
For \( m \) to be a natural number, the term \( \frac{(n - 7)(n + 5)}{n - 4} \) must also be a natural number. This implies that \( n - 4 \) must divide \( (n - 7)(n + 5) \).

### Step 5: Set conditions for \( n \)
To ensure that \( m \) is a natural number, \( n - 4 \) must divide the constant term \( -23 \) (which comes from substituting \( n = 4 \) into the numerator). Therefore, we need to find values of \( n \) such that \( n - 4 \) is a divisor of \( -23 \).

The divisors of \( 23 \) are \( \pm 1, \pm 23 \). Hence, we can set:
1. \( n - 4 = 1 \) → \( n = 5 \)
2. \( n - 4 = 23 \) → \( n = 27 \)

### Step 6: Calculate \( m \) for valid \( n \)
Now we will calculate \( m \) for both values of \( n \):

1. For \( n = 5 \):
\[ m = \frac{(5 - 7)(5 + 5)}{5 - 4} = \frac{(-2)(10)}{1} = -20 \quad \text{(not a natural number)} \]

2. For \( n = 27 \):
\[ m = \frac{(27 - 7)(27 + 5)}{27 - 4} = \frac{(20)(32)}{23} = \frac{640}{23} \quad \text{(not an integer)} \]

### Step 7: Check other values of \( n \)
Since \( n - 4 \) must be a divisor of \( -23 \), we can also check \( n - 4 = -1 \) and \( n - 4 = -23 \), leading to:
1. \( n - 4 = -1 \) → \( n = 3 \)
2. \( n - 4 = -23 \) → \( n = -19 \) (not a natural number)

For \( n = 3 \):
\[ m = \frac{(3 - 7)(3 + 5)}{3 - 4} = \frac{(-4)(8)}{-1} = 32 \]

### Final Result
Thus, the only valid solution is:
\[ m = 32 \quad \text{and} \quad n = 3 \]