For $\frac{n+18}{n}$ to be an integer, $n$ must divide $n+18$.
In other words, $n$ must be a divisor of 18.
The positive divisors of 18 are 1, 2, 3, 6, 9, and 18.
So, the sum of all positive integer values of $n$ is $1+2+3+6+9+18=\boxed{39}$.
The answer is: 39
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer?

### Answer:
