E. G-C-D, Unlucky!
math; number theory
Two arrays \(p\) and \(s\) of length \(n\) are given, where \(p\) is the prefix GCD\(^{\text{∗}}\) of some array \(a\), and \(s\) is the suffix GCD of the same array \(a\). In other words, if the array \(a\) existed, then for each \(1 \le i \le n\), the following equalities would hold both: \(p_i = \gcd(a_1, a_2, \dots, a_i)\) \(s_i = \gcd(a_i, a_{i+1}, \dots, a_n)\). Determine whether there exists such an array \(a\) for which the given arrays \(p\) and \(s\) can be obtained.\(^{\text{∗}}\)\(\gcd(x, y)\) denotes the greatest common divisor (GCD) of integers \(x\) and \(y\).
The first line contains an integer \(t\) (\(1 \le t \le 10^4\)) — the number of test cases.Each test case consists of three lines:The first line of each test case contains a single integer \(n\) (\(1 \leq n \leq 10^5\)) — the length of the array.The second line of each test case contains \(n\) integers \(p_1, p_2, \dots, p_n\) (\(1 \leq p_i \le 10^9\)) — the elements of the array.The third line of each test case contains \(n\) integers \(s_1, s_2, \dots, s_n\) (\(1 \leq s_i \le 10^9\)) — the elements of the array.It is guaranteed that the sum of all \(n\) across all test cases does not exceed \(10^5\).
For each test case, output ""Yes"" (without quotes) if there exists an array \(a\) for which the given arrays \(p\) and \(s\) can be obtained, and ""No"" (without quotes) otherwise.You may output each letter in any case (lowercase or uppercase). For example, the strings ""yEs"", ""yes"", ""Yes"", and ""YES"" will be accepted as a positive answer.
For the first test case, a possible array is: [\(72,\ 24,\ 3,\ 6,\ 12,\ 144\)].For the second test case, it can be shown that such arrays do not exist.For the third test case, there exists an array: [\(125,\ 125,\ 125,\ 25,\ 75\)].