As $n$ ranges over the positive integers, what is the sum of all possible values of the greatest common divisor of $3n+4$ and $n$?

Applying Euclidean's algorithm, we get:

\(\gcd(3n+4, n) \\=\gcd(3n+4-3n, n)\\=\gcd(4, n)\)

The factors of 4 are 1, 2, and 4. Sum them up and you'll get the answer.