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 can rewrite the expression as follows:
\[ \frac{15n^2 + 8n + 6}{n} = \frac{15n^2}{n} + \frac{8n}{n} + \frac{6}{n} \]
This simplifies to:
\[ 15n + 8 + \frac{6}{n} \]

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

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

4. **Count the Natural Numbers**:
The natural numbers \( n \) that satisfy the condition are \( 1, 2, 3, \) and \( 6 \). Thus, there are a total of 4 natural numbers.

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

---