E. Common Generator
brute force; constructive algorithms; math; number theory
For two integers \(x\) and \(y\) (\(x,y\ge 2\)), we will say that \(x\) is a generator of \(y\) if and only if \(x\) can be transformed to \(y\) by performing the following operation some number of times (possibly zero): Choose a divisor \(d\) (\(d\ge 2\)) of \(x\), then increase \(x\) by \(d\). For example, \(3\) is a generator of \(8\) since we can perform the following operations: \(3 \xrightarrow{d = 3} 6 \xrightarrow{d = 2} 8\); \(4\) is a generator of \(10\) since we can perform the following operations: \(4 \xrightarrow{d = 4} 8 \xrightarrow{d = 2} 10\); \(5\) is not a generator of \(6\) since we cannot transform \(5\) into \(6\) with the operation above. Now, Kevin gives you an array \(a\) consisting of \(n\) pairwise distinct integers (\(a_i\ge 2\)).You have to find an integer \(x\ge 2\) such that for each \(1\le i\le n\), \(x\) is a generator of \(a_i\), or determine that such an integer does not exist.
Each test contains multiple test cases. The first line of the input contains a single integer \(t\) (\(1\le t\le 10^4\)) — the number of test cases. The description of test cases follows.The first line of each test case contains a single integer \(n\) (\(1\le n\le 10^5\)) — the length of the array \(a\).The second line contains \(n\) integers \(a_1,a_2,\ldots,a_n\) (\(2\le a_i\le 4\cdot 10^5\)) — the elements in the array \(a\). It is guaranteed that the elements are pairwise distinct.It is guaranteed that the sum of \(n\) over all test cases does not exceed \(10^5\).
For each test case, output a single integer \(x\) — the integer you found. Print \(-1\) if there does not exist a valid \(x\).If there are multiple answers, you may output any of them.
In the first test case, for \(x=2\): \(2\) is a generator of \(8\), since we can perform the following operations: \(2 \xrightarrow{d = 2} 4 \xrightarrow{d = 4} 8\); \(2\) is a generator of \(9\), since we can perform the following operations: \(2 \xrightarrow{d = 2} 4 \xrightarrow{d = 2} 6 \xrightarrow{d = 3} 9\). \(2\) is a generator of \(10\), since we can perform the following operations: \(2 \xrightarrow{d = 2} 4 \xrightarrow{d = 2} 6 \xrightarrow{d = 2} 8 \xrightarrow{d = 2} 10\). In the second test case, it can be proven that it is impossible to find a common generator of the four integers.