CF1909B.Make Almost Equal With Mod

xi - Solar Storm

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You are given an array $a_1, a_2, \dots, a_n$ of distinct positive integers. You have to do the following operation exactly once:

• choose a positive integer $k$ ;
• for each $i$ from $1$ to $n$ , replace $a_i$ with $a_i \text{ mod } k^\dagger$ .

Find a value of $k$ such that $1 \leq k \leq 10^{18}$ and the array $a_1, a_2, \dots, a_n$ contains exactly $2$ distinct values at the end of the operation. It can be shown that, under the constraints of the problem, at least one such $k$ always exists. If there are multiple solutions, you can print any of them.

$^\dagger$ $a \text{ mod } b$ denotes the remainder after dividing $a$ by $b$ . For example:

• $7 \text{ mod } 3=1$ since $7 = 3 \cdot 2 + 1$
• $15 \text{ mod } 4=3$ since $15 = 4 \cdot 3 + 3$
• $21 \text{ mod } 1=0$ since $21 = 21 \cdot 1 + 0$

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 500$ ). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $2 \le n \le 100$ ) — the length of the array $a$ .

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le 10^{17}$ ) — the initial state of the array. It is guaranteed that all the $a_i$ are distinct.

Note that there are no constraints on the sum of $n$ over all test cases.

For each test case, output a single integer: a value of $k$ ( $1 \leq k \leq 10^{18}$ ) such that the array $a_1, a_2, \dots, a_n$ contains exactly $2$ distinct values at the end of the operation.

In the first test case, you can choose $k = 7$ . The array becomes $[8 \text{ mod } 7, 15 \text{ mod } 7, 22 \text{ mod } 7, 30 \text{ mod } 7] = [1, 1, 1, 2]$ , which contains exactly $2$ distinct values ( $\{1, 2\}$ ).

In the second test case, you can choose $k = 30$ . The array becomes $[0, 0, 8, 0, 8]$ , which contains exactly $2$ distinct values ( $\{0, 8\}$ ). Note that choosing $k = 10$ would also be a valid solution.

In the last test case, you can choose $k = 10^{18}$ . The array becomes $[2, 1]$ , which contains exactly $2$ distinct values ( $\{1, 2\}$ ). Note that choosing $k = 10^{18} + 1$ would not be valid, because $1 \leq k \leq 10^{18}$ must be true.