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.StringTokenizer;
  
  public class PaintArray {
      long gcd(long a, long b) {
          if(b==0) return a;
          return gcd(b, a % b);
      }
      long gcdAll(long arr[], int start) {
          long g = arr[start];
          if(arr.length>2) {
              for (int j = start + 2; j < arr.length; j += 2) {
                  g = gcd(arr[j], g);
              }
          }
          return g;
      }
      boolean checkAll(long arr[], int start, long g) {
          boolean good = true;
          for (int j = start; j < arr.length; j+=2) {
              if (arr[j] % g == 0) {
                  good = false;
                  break;
              }
          }
          return good;
      }
      void solve() {
          int t = scanner.nextInt();
          for (int i = 0; i < t; i++) {
              int n = scanner.nextInt();
              long arr[] = scanner.nextLongArr();
              long g = gcdAll(arr, 0);
              if (checkAll(arr, 1, g)) {
                  out.println(g);
              } else {
                  g = gcdAll(arr, 1);
                  if (checkAll(arr, 0, g)) {
                      out.println(g);
                  } else {
                      out.println('0');
                  }
              }
          }
          out.close();
      }
  
      InputStream sin;
      FScanner scanner;
      PrintWriter out;
      PaintArray() {
          this(System.in, System.out);
      }
      PaintArray(InputStream sin, OutputStream sout) {
          this.sin = sin;
          scanner = new FScanner(sin);
          out = new PrintWriter(new BufferedOutputStream(sout));
      }
      public static void main(String[] args) {
          new PaintArray().solve();
      }
      static class FScanner implements AutoCloseable {
          private BufferedReader bufferedReader;
          private StringTokenizer tokenizer;
          public FScanner(InputStream inputStream) {
              this.bufferedReader = new BufferedReader(new InputStreamReader(inputStream));
          }
          public int nextInt() {
              return Integer.parseInt(next());
          }
          public int[] nextIntArr() {
              String line = nextLine();
              String[] arr = line.split(" ");
              int result[] = new int[arr.length];
              for (int i = 0; i < arr.length; i++) {
                  result[i] = Integer.parseInt(arr[i]);
              }
              return result;
          }
          public long[] nextLongArr() {
              String line = nextLine();
              String[] arr = line.split(" ");
              long result[] = new long[arr.length];
              for (int i = 0; i < arr.length; i++) {
                  result[i] = Long.parseLong(arr[i]);
              }
              return result;
          }
          public Double nextDouble() {
              return Double.parseDouble(next());
          }
          public String next() {
              skipUntilTokenOrEnd();
              return tokenizer.nextToken();
          }
          private void skipUntilTokenOrEnd() {
              while(tokenizer==null || !tokenizer.hasMoreElements()) {
                  try {
                      String line = bufferedReader.readLine();
                      if (line == null) {
                          throw new RuntimeException("No more tokens");
                      }
                      tokenizer = new StringTokenizer(line);
                  } catch (IOException e) {
                      throw new RuntimeException(e);
                  }
              }
          }
          @Override
          public void close() {
              try {
                  bufferedReader.close();
              } catch (IOException e) {
                  throw new RuntimeException(e);
              }
          }
          public String nextLine() {
              skipUntilTokenOrEnd();
              StringBuilder sb = new StringBuilder();
              while(tokenizer.hasMoreTokens()) {
                  sb.append(tokenizer.nextToken());
                  sb.append(' ');
              }
              sb.delete(sb.length() - 1, sb.length());
              return sb.toString();
          }
      }
  }
  