description: You are given an array a consisting of n positive integers. You have to choose a positive integer d and paint all elements into two colors. All elements which are divisible by d will be painted red, and all other elements will be painted blue.
  
  The coloring is called beautiful if there are no pairs of adjacent elements with the same color in the array. Your task is to find any value of d which yields a beautiful coloring, or report that it is impossible.
  
  Input
  
  The first line contains a single integer t (1 ≤ t ≤ 1000) — the number of testcases.
  
  The first line of each testcase contains one integer n (2 ≤ n ≤ 100) — the number of elements of the array.
  
  The second line of each testcase contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^{18}).
  
  Output
  
  For each testcase print a single integer. If there is no such value of d that yields a beautiful coloring, print 0. Otherwise, print any suitable value of d (1 ≤ d ≤ 10^{18}).
  
  Example
  
  Input
  
  
  5
  5
  1 2 3 4 5
  3
  10 5 15
  3
  100 10 200
  10
  9 8 2 6 6 2 8 6 5 4
  2
  1 3
  
  
  Output
  
  
  2
  0
  100
  0
  3
solution: import os
  import sys
  from io import BytesIO, IOBase
  import math
  from decimal import Decimal
  from decimal import *
  from collections import defaultdict, deque
  import heapq
  import bisect
  from decimal import Decimal
  getcontext().prec = 25
  abcd='abcdefghijklmnopqrstuvwxyz'
  ABCD='ABCDEFGHIJKLMNOPQRSTUVWXYZ'
  months = {1:31, 2:28,3:31, 4:30, 5:31, 6:30, 7:31, 8:31, 9:30, 10:31, 11:30, 12:31}
  MOD = 1000000007
  BUFSIZE = 8192
  # sys.setrecursionlimit(10**6)
  from bisect import bisect_left, bisect_right
  class FastIO(IOBase):
      newlines = 0
  
      def __init__(self, file):
          self._fd = file.fileno()
          self.buffer = BytesIO()
          self.writable = "x" in file.mode or "r" not in file.mode
          self.write = self.buffer.write if self.writable else None
  
      def read(self):
          while True:
              b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
              if not b:
                  break
              ptr = self.buffer.tell()
              self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
          self.newlines = 0
          return self.buffer.read()
  
      def readline(self):
          while self.newlines == 0:
              b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
              self.newlines = b.count(b"\n") + (not b)
              ptr = self.buffer.tell()
              self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
          self.newlines -= 1
          return self.buffer.readline()
  
      def flush(self):
          if self.writable:
              os.write(self._fd, self.buffer.getvalue())
              self.buffer.truncate(0), self.buffer.seek(0)
  
  
  class IOWrapper(IOBase):
      def __init__(self, file):
          self.buffer = FastIO(file)
          self.flush = self.buffer.flush
          self.writable = self.buffer.writable
          self.write = lambda s: self.buffer.write(s.encode("ascii"))
          self.read = lambda: self.buffer.read().decode("ascii")
          self.readline = lambda: self.buffer.readline().decode("ascii")
  
  sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
  input = lambda: sys.stdin.readline().rstrip("\r\n")
  
  # for _ in range(int(input())):
  # map(int, input().split(" "))
  # list(map(int, input().split(" ")))
  
  
  for _ in range(int(input())):
      n = int(input())
      l = list(map(int, input().split(" ")))
      gcd = 0
      gcd_2 = 0
      for i in range(n):
          if i%2:
              gcd = math.gcd(gcd, l[i])
          else:
              gcd_2 = math.gcd(gcd_2, l[i])
  
      if gcd_2==gcd:
          print(0)
          continue
      check = True
      if gcd!=1:
          for i in range(n):
              if i%2==0:
                  if l[i]%gcd==0:
                      check=False
      if check and gcd!=1:
          print(gcd)
          continue
      check = True
      if gcd_2!=1:
          for i in range(n):
              if i%2:
                  if l[i]%gcd_2==0:
                      check=False
      if check and gcd_2!=1:
          print(gcd_2)
          continue
      print(0)