CF2124C Subset Multiplication

Alice has an array $ a $ , consisting of $ n $ positive integers. The array satisfies the beautiful property that $ a_i $ divides $ a_{i+1} $ for each $ 1 \leq i \leq n - 1 $ . Bob sees Alice's beautiful array and is jealous. To sabotage her, Bob first creates an array $ b $ of size $ n $ such that $ b_i=a_i $ for each $ 1 \leq i \leq n $ . Then, he chooses a positive integer $ x $ and multiplies some (possibly none, possibly all) elements in $ b $ by $ x $ . Formally, he chooses a (possibly-empty) subset $ S\subseteq\{1,2,\ldots,n\} $ , and for each $ i\in S $ , he sets $ b_i:=b_i\cdot x $ . You are given an array $ b $ , but you don't know array $ a $ and the chosen number $ x $ . Please output any integer $ x $ that Bob could choose, so that multiplying some subset of elements of the correct array $ a $ by $ x $ would result in array $ b $ . It is guaranteed that the answer exists. If there are multiple possible integers, you can output any of them.

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 2\cdot10^5 $ ). The description of the test cases follows. The first line of each test case contains an integer $ n $ ( $ 2 \leq n \leq 6\cdot10^5 $ ) — the length of the array $ b $ . The second line of each test case contains $ n $ integers $ b_1,b_2,\ldots,b_n $ ( $ 1 \leq b_i \leq 10^9 $ ) — denoting the array $ b $ . It is guaranteed the array $ b $ can be obtained from some beautiful array $ a $ and some positive integer $ x $ as described in the statements. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 6\cdot 10^5 $ .

For each test case, output any possible value of $ x $ ( $ 1 \leq x \leq 10^9 $ ) on a new line. It is guaranteed at least one value of $ x $ exists.

In the first test case, it is possible Bob selected $ x=343 $ and $ S=\{\} $ (meaning he did not change the array $ a $ at all). In the third test case, it is possible Bob selected $ x=4 $ and $ S=\{1,2\} $ , meaning he multiplied both $ b_1 $ and $ b_2 $ by $ 4 $ . The original array was $ \{1,2,4,8\} $ , which satisfies the required property.