What is the maximum integral value of m, if `n<0` and m = n+(285)/n ?

Options

A. -34
B. -286
C. 34
D. -33

Answer

To find the maximum integral value of \( m \) given that \( n < 0 \) and \( m = n + \frac{285}{n} \), we can follow these steps:

### Step 1: Rewrite the expression for \( m \)
We start with the equation:
\[ m = n + \frac{285}{n} \]

### Step 2: Find the derivative to locate critical points
To maximize \( m \), we can take the derivative of \( m \) with respect to \( n \) and set it to zero. However, since \( n \) must be negative, we can also analyze the function directly by substituting negative factors of 285.

### Step 3: Identify factors of 285
The factors of 285 are:
- \( -1, -3, -5, -15, -19, -57, -95, -285 \)

### Step 4: Calculate \( m \) for each negative factor
We will calculate \( m \) for each negative factor of 285:

1. **For \( n = -1 \)**:
\[ m = -1 + \frac{285}{-1} = -1 - 285 = -286 \]

2. **For \( n = -3 \)**:
\[ m = -3 + \frac{285}{-3} = -3 - 95 = -98 \]

3. **For \( n = -5 \)**:
\[ m = -5 + \frac{285}{-5} = -5 - 57 = -62 \]

4. **For \( n = -15 \)**:
\[ m = -15 + \frac{285}{-15} = -15 - 19 = -34 \]

5. **For \( n = -19 \)**:
\[ m = -19 + \frac{285}{-19} = -19 - 15 = -34 \]

6. **For \( n = -57 \)**:
\[ m = -57 + \frac{285}{-57} = -57 - 5 = -62 \]

7. **For \( n = -95 \)**:
\[ m = -95 + \frac{285}{-95} = -95 - 3 = -98 \]

8. **For \( n = -285 \)**:
\[ m = -285 + \frac{285}{-285} = -285 - 1 = -286 \]

### Step 5: Identify the maximum value of \( m \)
From the calculations above, the values of \( m \) for the negative factors of 285 are:
- \( -286, -98, -62, -34, -34, -62, -98, -286 \)

The maximum integral value of \( m \) is:
\[ \text{Maximum } m = -34 \]

### Conclusion
Thus, the maximum integral value of \( m \) is:
\[ \boxed{-34} \]