If N is any four digit number say `x_1, x_2, x_3, x_4`, then the maximum value ofis equal to `N/(x_1+x_2+x_3+x_4)` is equal to

Options

A. 1000
B. `1111/4`
C. 800
D. none of these

Answer

To solve the problem of finding the maximum value of \( \frac{N}{x_1 + x_2 + x_3 + x_4} \) where \( N \) is a four-digit number represented by its digits \( x_1, x_2, x_3, x_4 \), we can follow these steps:

### Step 1: Understand the constraints
A four-digit number \( N \) can range from 1000 to 9999. The digits \( x_1, x_2, x_3, x_4 \) represent the thousands, hundreds, tens, and units place respectively.

### Step 2: Express \( N \) in terms of its digits
The four-digit number \( N \) can be expressed as:
\[ N = 1000x_1 + 100x_2 + 10x_3 + x_4 \]

### Step 3: Calculate the sum of the digits
The sum of the digits is:
\[ S = x_1 + x_2 + x_3 + x_4 \]

### Step 4: Set up the expression to maximize
We need to maximize the expression:
\[ \frac{N}{S} = \frac{1000x_1 + 100x_2 + 10x_3 + x_4}{x_1 + x_2 + x_3 + x_4} \]

### Step 5: Analyze extreme cases
1. **Minimum case**: If \( N = 1000 \) (i.e., \( x_1 = 1, x_2 = 0, x_3 = 0, x_4 = 0 \)):
\[ S = 1 + 0 + 0 + 0 = 1 \quad \Rightarrow \quad \frac{1000}{1} = 1000 \]

2. **Maximum case**: If \( N = 9999 \) (i.e., \( x_1 = 9, x_2 = 9, x_3 = 9, x_4 = 9 \)):
\[ S = 9 + 9 + 9 + 9 = 36 \quad \Rightarrow \quad \frac{9999}{36} \approx 277.75 \]

### Step 6: Check other combinations
To ensure we have the maximum, we can check other combinations of digits. For example:
- If \( N = 2000 \) (i.e., \( x_1 = 2, x_2 = 0, x_3 = 0, x_4 = 0 \)):
\[ S = 2 \quad \Rightarrow \quad \frac{2000}{2} = 1000 \]
- If \( N = 3000 \) (i.e., \( x_1 = 3, x_2 = 0, x_3 = 0, x_4 = 0 \)):
\[ S = 3 \quad \Rightarrow \quad \frac{3000}{3} = 1000 \]
- Continuing this way, we find that for \( N = 4000, 5000, \ldots, 9000 \), the value remains 1000.

### Conclusion
The maximum value of \( \frac{N}{x_1 + x_2 + x_3 + x_4} \) is \( 1000 \) when \( N \) is of the form \( 1000, 2000, 3000, \ldots, 9000 \).

### Final Answer
\[ \text{Maximum value} = 1000 \]