#include <bits/stdc++.h>
using namespace std;
int main() {
  ios_base::sync_with_stdio(false);
  cin.tie(NULL);
  int t;
  cin >> t;
  while (t--) {
    int n;
    cin >> n;
    long long int arr[n];
    for (int i = 0; i < n; i++) {
      long long int a;
      cin >> a;
      arr[i] = a;
    }
    sort(arr, arr + n);
    vector<long long int> v;
    if (n % 2 == 0) {
      for (int i = 0; i < n / 2; i++) {
        if (v.size() == 0) {
          v.push_back(arr[i] * arr[n - i - 1]);
        } else {
          if (arr[i] * arr[n - i - 1] != v[v.size() - 1]) {
            v.push_back(arr[i] * arr[n - i - 1]);
          }
        }
      }
    } else {
      for (int i = 0; i <= n / 2; i++) {
        if (i == n / 2) {
          if (v.size() == 0) {
            v.push_back(arr[i] * arr[i]);
          } else {
            if (arr[i] * arr[i] != v[v.size() - 1]) {
              v.push_back(arr[i] * arr[i]);
            }
          }
        } else {
          if (v.size() == 0) {
            v.push_back(arr[i] * arr[n - i - 1]);
          } else {
            if (arr[i] * arr[n - i - 1] != v[v.size() - 1]) {
              v.push_back(arr[i] * arr[n - i - 1]);
            }
          }
        }
      }
    }
    long long int ans = 0;
    if (v.size() > 1) {
      ans = -1;
    } else {
      int count = 0;
      for (long long int i = 2; i * i <= v[0]; i++) {
        if (v[0] % i == 0) {
          if (i * i == v[0]) {
            count++;
          } else {
            count += 2;
          }
        }
      }
      if (count == n) {
        ans = v[0];
      } else {
        ans = -1;
      }
    }
    cout << ans << endl;
  }
}
