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 java.io.*;
  import java.util.*;
  
  public class PaintTheArray implements Runnable {
  
     void solve() {
        int t = ri();
        outer: while (t-- > 0) {
           int len = ri();
           long[] arr = lArr(len);
           for (int i = 0; i < len; i++) {
              arr[i] = rl();
           }
           long gcd1 = -1, gcd2 = -1;
           for (int i = 0; i < len; i += 2) {
              if (gcd1 == -1) {
                 gcd1 = arr[i];
              } else {
                 gcd1 = find(gcd1, arr[i]);
              }
           }
           for (int i = 1; i < len; i += 2) {
              if (gcd2 == -1) {
                 gcd2 = arr[i];
              } else {
                 gcd2 = find(gcd2, arr[i]);
              }
           }
  
           // check for both gcds
           boolean isFound = true;
           for (int i = 1; i < len; i+=2) {
              if (doDivide(gcd1, arr[i])) {
                 isFound = false;
                 break;
              }
           }
           if (isFound) {
              println(gcd1);
              continue outer;
           }
           isFound = true;
  
           for (int i = 0; i < len; i+=2) {
              if (doDivide(gcd2, arr[i])) {
                 isFound = false;
                 break;
              }
           }
           if (isFound) {
              println(gcd2);
              continue outer;
           }
           println(0);
        }
     }
  
     boolean doDivide(long a, long b) {
        return (((b / a) * a) == b);
     }
  
     long find(long a, long b) {
        if (a > b)
           a = a ^ b ^ (b = a);
        return gcd(a, b);
     }
  
     long gcd(long a, long b) {
        return ((a == 0) ? b : gcd(b % a, a));
     }
  
     /************************************************************************************************************************************************/
     public static void main(String[] args) throws IOException {
        new Thread(null, new PaintTheArray(), "1").start();
     }
  
     static PrintWriter out = new PrintWriter(System.out);
     static Reader read = new Reader();
     static StringBuilder sbr = new StringBuilder();
     static int mod = (int) 1e9 + 7;
     static int dmax = Integer.MAX_VALUE;
     static long lmax = Long.MAX_VALUE;
     static int dmin = Integer.MIN_VALUE;
     static long lmin = Long.MIN_VALUE;
     static int[] dprimes = new int[] { 1, 11, 101, 1087, 99991, 100001, 1000003, 15485863, 999999937 };
  
     @Override
     public void run() {
        solve();
        out.close();
     }
  
     static class Reader {
        private byte[] buf = new byte[1024];
        private int index;
        private InputStream in;
        private int total;
  
        public Reader() {
           in = System.in;
        }
  
        public int scan() throws IOException {
           if (total < 0)
              throw new InputMismatchException();
           if (index >= total) {
              index = 0;
              total = in.read(buf);
              if (total <= 0)
                 return -1;
           }
           return buf[index++];
        }
  
        public int intNext() throws IOException {
           int integer = 0;
           int n = scan();
           while (isWhiteSpace(n))
              n = scan();
           int neg = 1;
           if (n == '-') {
              neg = -1;
              n = scan();
           }
           while (!isWhiteSpace(n)) {
              if (n >= '0' && n <= '9') {
                 integer *= 10;
                 integer += n - '0';
                 n = scan();
              } else
                 throw new InputMismatchException();
           }
           return neg * integer;
        }
  
        public double doubleNext() throws IOException {
           double doub = 0;
           int n = scan();
           while (isWhiteSpace(n))
              n = scan();
           int neg = 1;
           if (n == '-') {
              neg = -1;
              n = scan();
           }
           while (!isWhiteSpace(n) && n != '.') {
              if (n >= '0' && n <= '9') {
                 doub *= 10;
                 doub += n - '0';
                 n = scan();
              } else
                 throw new InputMismatchException();
           }
           if (n == '.') {
              n = scan();
              double temp = 1;
              while (!isWhiteSpace(n)) {
                 if (n >= '0' && n <= '9') {
                    temp /= 10;
                    doub += (n - '0') * temp;
                    n = scan();
                 } else
                    throw new InputMismatchException();
              }
           }
           return doub * neg;
        }
  
        public String read() throws IOException {
           StringBuilder sb = new StringBuilder();
           int n = scan();
           while (isWhiteSpace(n))
              n = scan();
           while (!isWhiteSpace(n)) {
              sb.append((char) n);
              n = scan();
           }
           return sb.toString();
        }
  
        public String rline() throws IOException {
           StringBuilder sb = new StringBuilder();
           int n = scan();
           while (n != '\n') {
              sb.append((char) n);
              n = scan();
           }
           return sb.toString();
        }
  
        private boolean isWhiteSpace(int n) {
           if (n == ' ' || n == '\n' || n == '\r' || n == '\t' || n == -1)
              return true;
           return false;
        }
     }
  
     static void print(Object object) {
        out.print(object);
     }
  
     static void println(Object object) {
        out.println(object);
     }
  
     static int[] iArr(int len) {
        return new int[len];
     }
  
     static long[] lArr(int len) {
        return new long[len];
     }
  
     static long min(long a, long b) {
        return Math.min(a, b);
     }
  
     static int min(int a, int b) {
        return Math.min(a, b);
     }
  
     static long max(Long a, Long b) {
        return Math.max(a, b);
     }
  
     static int max(int a, int b) {
        return Math.max(a, b);
     }
  
     static int ri() {
        try {
           return read.intNext();
        } catch (IOException e) {
           e.printStackTrace();
        }
        System.exit(2);
        return -1;
     }
  
     static long rl() {
        try {
           return Long.parseLong(read.read());
        } catch (IOException e) {
           e.printStackTrace();
        }
        System.exit(2);
        return -1;
     }
  
     static String rs() {
        try {
           return read.read();
        } catch (IOException e) {
           e.printStackTrace();
        }
        System.exit(2);
        return "";
     }
  
     static String rline() {
        try {
           return read.rline();
        } catch (IOException e) {
           e.printStackTrace();
        }
        System.exit(2);
        return "";
     }
  
     static char rc() {
        return rs().charAt(0);
     }
  
     static double rd() {
        try {
           return read.doubleNext();
        } catch (IOException e) {
           e.printStackTrace();
        }
        System.exit(2);
        return -1;
     }
  }
  