CF1780B.GCD Partition

While at Kira's house, Josuke saw a piece of paper on the table with a task written on it.

The task sounded as follows. There is an array $a$ of length $n$ . On this array, do the following:

• select an integer $k > 1$ ;
• split the array into $k$ subsegments $^\dagger$ ;
• calculate the sum in each of $k$ subsegments and write these sums to another array $b$ (where the sum of the subsegment $(l, r)$ is ${\sum_{j = l}^{r}a_j}$ );
• the final score of such a split will be $\gcd(b_1, b_2, \ldots, b_k)^\ddagger$ .

The task is to find such a partition that the score is maximum possible. Josuke is interested in this task but is not strong in computer science. Help him to find the maximum possible score.

$^\dagger$ A division of an array into $k$ subsegments is $k$ pairs of numbers $(l_1, r_1), (l_2, r_2), \ldots, (l_k, r_k)$ such that $l_i \le r_i$ and for every $1 \le j \le k - 1$ $l_{j + 1} = r_j + 1$ , also $l_1 = 1$ and $r_k = n$ . These pairs represent the subsegments.

$^\ddagger$ $\gcd(b_1, b_2, \ldots, b_k)$ stands for the greatest common divisor (GCD) of the array $b$ .

The first line contains a single number $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

For each test case, the first line contains one integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ) — the length of the array $a$ .

The second line contains $n$ integers $a_1, a_2, a_3, \ldots, a_n$ ( $1 \le a_i \le 10^9 $ ) — the array $a$ itself.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case print a single integer — the maximum score for the optimal partition.

In the first test case, you can choose $k = 2$ and split the array into subsegments $(1, 2)$ and $(3, 4)$ .

Then the score of such a partition will be equal to $\gcd(a_1 + a_2, a_3 + a_4) = \gcd(2 + 2, 1 + 3) = \gcd(4, 4) = 4$ .

In the fourth test case, you can choose $k = 3$ and split the array into subsegments $(1, 2), (3, 5), (6, 6)$ .

The split score is $\gcd(1 + 2, 1 + 1 + 1, 3) = 3$ .