Let $m$ and $n$ be positive integers. We aim to find the minimum positive integer $N$ which satisfies the following condition: If there exists a set $S$ of integers that contains a complete residue system modulo $m$ such that $|S| = N$, then there exists a nonempty set $A \subseteq S$ so that $n \mid \sum_{x \in A} x$.

First, let $d = \gcd(m, n)$, and write $m = ad$ and $n = bd$. The answer depends on the relationship between $bd$ and $\frac{ad(d+1)}{2}$.

The minimum positive integer $N$ is given by:
$$N = \begin{cases} 1 & \text{if } bd \leq \frac{ad(d+1)}{2}, \\ bd - \frac{ad(d-1)}{2} & \text{otherwise}. \end{cases}$$

The answer is: \boxed{\begin{cases}
1 & \text{if } bd \leq \frac{ad(d+1)}{2}, \\
bd - \frac{ad(d-1)}{2} & \text{otherwise}.
\end{cases}}.