For every natural number n, the number of people in a village is (n+150) and each family of this village has (n+6) members. If in each village of this region the number of families is distinct, find the maximum number of villages in the region

Options

A. 1. 8
B. 2. 9
C. 3. 14
D. 4. none of the these

Answer

To solve the problem step by step, we need to determine the number of families in a village based on the given formulas and then find the maximum number of distinct families for different values of \( n \).

### Step 1: Understand the problem
We are given:
- The number of people in a village: \( n + 150 \)
- The number of members in each family: \( n + 6 \)

We need to find the number of families in the village and ensure that this number is distinct for different natural numbers \( n \).

### Step 2: Calculate the number of families
The number of families in the village can be calculated using the formula:
\[ \text{Number of families} = \frac{\text{Total number of people}}{\text{Number of members in each family}} = \frac{n + 150}{n + 6} \]

### Step 3: Simplify the expression
To simplify the expression:
\[ \text{Number of families} = \frac{n + 150}{n + 6} \]
We can perform polynomial long division or simply rewrite it:
\[ \text{Number of families} = 1 + \frac{144}{n + 6} \]

### Step 4: Determine the distinct values
For the number of families to be distinct for different values of \( n \), the expression \( \frac{144}{n + 6} \) must yield different results for different \( n \).

### Step 5: Find the maximum number of villages
To find the maximum number of villages, we need to consider the values of \( n \) that give distinct integer results for the number of families.

1. The expression \( \frac{144}{n + 6} \) must be an integer. This means \( n + 6 \) must be a divisor of 144.
2. The divisors of 144 are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144.

### Step 6: Calculate possible values of \( n \)
For each divisor \( d \) of 144, we can find \( n \):
\[ n + 6 = d \implies n = d - 6 \]
We need \( n \) to be a natural number, so \( d \) must be greater than 6.

The divisors of 144 greater than 6 are:
- 8 (gives \( n = 2 \))
- 9 (gives \( n = 3 \))
- 12 (gives \( n = 6 \))
- 16 (gives \( n = 10 \))
- 18 (gives \( n = 12 \))
- 24 (gives \( n = 18 \))
- 36 (gives \( n = 30 \))
- 48 (gives \( n = 42 \))
- 72 (gives \( n = 66 \))
- 144 (gives \( n = 138 \))

### Step 7: Count the distinct values of \( n \)
The distinct values of \( n \) we found are:
- 2, 3, 6, 10, 12, 18, 30, 42, 66, 138

This gives us a total of 10 distinct values of \( n \).

### Conclusion
Thus, the maximum number of villages in the region, where each village has a distinct number of families, is **10**.