2019: [USACO 2024 January Contest Silver] Problem 3. Cowlendar

Bessie has woken up on a strange planet. In this planet, there are $N$ ($1\le N\le {10}^4$) months, with $a_1,\dots ,a_N$ days, respectively ($1\le a_i\le 4\cdot {10}^9$, all $a_i$ are integers). In addition, on the planet, there are also weeks, where each week is $L$ days, with $L$ being a positive integer. Interestingly, Bessie knows the following:

• For the correct $L$, each month is at least $4$ weeks long.
• For the correct $L$, there are at most $3$ distinct values of $a_{i}modL$.

Unfortunately, Bessie has forgotten what $L$ is! Help her by printing the sum of all possible values of $L$.

Note that the large size of integers involved in this problem may require the use of 64-bit integer data types (e.g., a "long long" in C/C++).

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