[Number Theory]
Problem: Suppose that $n$ is a positive integer and that the value of $\frac{n^{2}+n+15}{n}$ is an integer. Determine all possible values of $n$.

Solution: First, we see that $\frac{n^{2}+n+15}{n}=\frac{n^{2}}{n}+\frac{n}{n}+\frac{15}{n}=n+1+\frac{15}{n}$.

This means that $\frac{n^{2}+n+15}{n}$ is an integer exactly when $n+1+\frac{15}{n}$ is an integer.

Since $n+1$ is an integer, then $\frac{n^{2}+n+15}{n}$ is an integer exactly when $\frac{15}{n}$ is an integer.

The expression $\frac{15}{n}$ is an integer exactly when $n$ is a divisor of 15 .

Since $n$ is a positive integer, then the possible values of $n$ are 1, 3, 5, and 15 .

Final answer: 1, 3, 5, 15