CF2258A Odd Eraser

Given an array $ a_1, a_2, \ldots, a_n $ , you can apply the following operation any number of times (possibly zero): - Choose an integer $ k \geq 1 $ such that $ 2k+1 \le m $ and $ 2k+1 $ indices, $ i_1, i_2, \ldots, i_{2k+1} $ ( $ 1 \le i_1 \lt i_2 \lt \ldots \lt i_{2k+1} \le m $ ), where $ m $ is the current length of the array. Then, remove the $ i_{k+1} $ -th element from the array. Note that after any operation, the length of the array is reduced by one, and the rest of the array is concatenated. Let $ b_1, b_2, \ldots, b_m $ be the remaining array after all operations. What is the maximum possible value of $ \gcd(b_1, b_2, \ldots, b_m) $ , where $ \gcd $ of an array of integers denotes the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) of them?

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 $ n $ ( $ 1 \le n \le 100 $ ), denoting the size of the array. The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \le a_i \le 10^9 $ ).

For each test case, output a single integer — the maximum possible value.

In the first test case, the given array is $ [2, 4, 6, 7, 8, 9, 10] $ . Choosing indices $ [1, 3, 4, 6, 7] $ results in the removal of $ a_4 = 7 $ and the array $ [2, 4, 6, 8, 9, 10] $ . Then, choosing indices $ [2, 5, 6] $ results in the removal of $ a_5 = 9 $ and the array $ [2, 4, 6, 8, 10] $ . You can't get an answer greater than $ 2 $ .