The number of natural numbers n for which `(15n^2+8n+6)/n` is a natural number is:

Answer

To solve the problem, we need to determine the number of natural numbers \( n \) for which the expression \( \frac{15n^2 + 8n + 6}{n} \) is a natural number.

### Step-by-step Solution:

1. **Rewrite the Expression**:
We start by simplifying the expression:
\[ \frac{15n^2 + 8n + 6}{n} = \frac{15n^2}{n} + \frac{8n}{n} + \frac{6}{n} = 15n + 8 + \frac{6}{n} \]

2. **Identify the Condition for Natural Numbers**:
For \( 15n + 8 + \frac{6}{n} \) to be a natural number, \( \frac{6}{n} \) must also be a natural number. This means that \( n \) must be a divisor of 6.

3. **Find the Divisors of 6**:
The natural divisors of 6 are:
- 1
- 2
- 3
- 6

4. **Check Each Divisor**:
We will check each divisor to ensure that \( 15n + 8 + \frac{6}{n} \) is indeed a natural number:
- For \( n = 1 \):
\[ 15(1) + 8 + \frac{6}{1} = 15 + 8 + 6 = 29 \quad (\text{natural number}) \]
- For \( n = 2 \):
\[ 15(2) + 8 + \frac{6}{2} = 30 + 8 + 3 = 41 \quad (\text{natural number}) \]
- For \( n = 3 \):
\[ 15(3) + 8 + \frac{6}{3} = 45 + 8 + 2 = 55 \quad (\text{natural number}) \]
- For \( n = 6 \):
\[ 15(6) + 8 + \frac{6}{6} = 90 + 8 + 1 = 99 \quad (\text{natural number}) \]

5. **Count the Valid Natural Numbers**:
All divisors \( n = 1, 2, 3, 6 \) yield natural numbers for the expression. Thus, the total count of natural numbers \( n \) for which \( \frac{15n^2 + 8n + 6}{n} \) is a natural number is 4.

### Final Answer:
The number of natural numbers \( n \) for which \( \frac{15n^2 + 8n + 6}{n} \) is a natural number is **4**.