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 $a_i\mathrm{mod}L$.

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++).

The possible values of $L$ are 1, 2, 3, 4, 5, 6, and 7. For example, $L=7$ is valid because each month is at least length $4\cdot 7=28$ days long, and each month is either 0, 2, or 3 mod 7.

The possible values of $L$ are 1, 2, 3, 4, 6, and 7. For example, $L=6$ is valid because each month is at least length $4\cdot 6=24$ days long, and each month is either 1, 4, or 5 mod 6.

Problem credits: Brandon Wang

Source: USA Computing Olympiad.

Input Format

Output Format

Sample Input 1

Sample Output 1

Sample Input 2

Sample Output 2

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