Problem 1320

1. **Understanding the function $f(S)$**:
- For a set $S$ of integers, $f(S)$ counts the number of integers $k$ in the range $0 \le k < 2019$ such that there exist $s_1, s_2 \in S$ with $s_1 - s_2 = k$.
- This means $f(S)$ essentially counts the distinct differences $s_1 - s_2$ modulo 2019.

2. Partitioning the positive integers:
- We need to partition the positive integers into $m$ nonempty sets $S_1, S_2, \ldots, S_m$ and minimize the sum $f(S_1) + f(S_2) + \cdots + f(S_m)$.

3. **Lower bound of $x_m$**:
- Consider the numbers $1, 2, \ldots, 2019$. If we partition these into $m$ sets, each set $S_i$ will have $a_i$ elements.
- The function $f(S_i)$ will be at least $a_i$ because the differences $s_1 - s_2$ for $s_1, s_2 \in S_i$ will cover at least $a_i$ distinct values.
- Therefore, $f(S_1) + f(S_2) + \cdots + f(S_m) \geq a_1 + a_2 + \cdots + a_m = 2019$.

4. Achieving the lower bound:
- To achieve $f(S_1) + f(S_2) + \cdots + f(S_m) = 2019$, each set $S_i$ must cover exactly $a_i$ distinct differences.
- This can be achieved if each $S_i$ is an arithmetic sequence with a common difference that divides 2019.

5. **Possible values of $r_i$**:
- The divisors of 2019 are 1, 3, 673, and 2019.
- If $S_i$ is an arithmetic sequence with common difference $r_i$, then $r_i$ must be one of these divisors.

6. Covering system:
- We need to find sets $S_i$ such that the sum of their reciprocals equals 1: $\sum \frac{1}{r_i} = 1$.
- Additionally, $\sum \lceil \frac{2019}{r_i} \rceil = 2019$.

7. Possible partitions:
- $r_i = 1$: This gives one set covering all integers.
- $r_i = 3$: We need three sets, each covering every third integer.
- $r_i = 673$: We need 673 sets, each covering every 673rd integer.
- $r_i = 2019$: We need 2019 sets, each covering every 2019th integer.

8. Counting the number of solutions:
- The solutions are of the form $2019 - 672n$ (for $n \leq 3$), $2019 - 2n$ (for $n \leq 673$), and 1.
- This gives 676 solutions.

The final answer is $\boxed{ 100M + N = 100 \times 2019 + 676 = 201900 + 676 = 202576 }$.