How many positive integers $n$ exist such that the quotient $\frac{2 n^{2}+4 n+18}{3 n+3}$ is an integer?
As

$$
\frac{2 n^{2}+4 n+18}{3 n+3}=\frac{2}{3}\left[\frac{\left(n^{2}+2 n+1\right)+8}{n+1}\right]=\frac{1}{3}\left(2 n+2+\frac{16}{n+1}\right)
$$

it follows that the expression in parentheses must be a multiple of 3 and, in particular, $n+1$ must divide 16. Thus, $n$ can be $1, 3, 7$ or 15. From the table below, in each of these four cases, that is, for $n$ equal to 1, 3, 7, or 15, the given quotient results in an integer.

| $n$ | $\frac{2 n^{2}+4 n+18}{3 n+3}$ |
| :---: | :---: |
| 1 | 4 |
| 3 | 4 |
| 7 | 6 |
| 15 | 11 |