#include using namespace std; const int p = 1e9 + 7; const int g = 5; struct PrimeFactor { static long long mul(long long x, long long y, long long mod) { long long m = x, s = 0; for (; y; y >>= 1, m <<= 1, m = m >= mod ? m - mod : m) if (y & 1) s += m, s = s >= mod ? s - mod : s; return s; } static long long powmod(long long x, long long p, long long mod) { long long s = 1, m = x % mod; for (; p; m = mul(m, m, mod), p >>= 1) if (p & 1) s = mul(s, m, mod); return s; } static bool miller_rabin(long long n, int s = 7) { static const long long wits[7] = { 2, 325, 9375, 28178, 450775, 9780504, 1795265022 }; auto witness = [&](long long a, long long n, long long u, int t) { long long x = powmod(a, u, n), nx; for (int i = 0; i < t; ++i, x = nx) { nx = mul(x, x, n); if (nx == 1 && x != 1 && x != n - 1) return true; } return x != 1; }; if (n < 2) return 0; if (~n & 1) return n == 2; long long u = n - 1, t = 0, a; for (; ~u & 1; u >>= 1) ++t; while (s--) if ((a = wits[s] % n) && witness(a, n, u, t)) return false; return true; } static long long pollard_rho(long long n) { auto f = [](long long x, long long n) { return mul(x, x, n) + 1; }; if (~n & 1) return 2; while (true) { long long x = rand() % (n - 1) + 1, y = 2, d = 1; for (int sz = 2; d == 1; y = x, sz <<= 1) for (int i = 0; i < sz && d <= 1; ++i) x = f(x, n), d = __gcd(abs(x - y), n); if (d && n - d) return d; } } static vector> factorize(long long n) { set factors; function rec = [&](long long x) { if (x == 1) return; if (miller_rabin(x)) return void(factors.emplace(x)); auto u = pollard_rho(x); rec(u); for (auto p : factors) while (x % p == 0) x /= p; rec(x); }; rec(n); vector> res; for (auto p : factors) { res.emplace_back(p, 0); for (; n % p == 0; n /= p) ++res.back().second; } return res; } }; tuple ext_gcd(int a, int b) { if (!b) return {1, 0, a}; int x, y, g; tie(x, y, g) = ext_gcd(b, a % b); return {y, x - a / b * y, g}; } const int magic = 1000; map mp; int x; int discrete_log(int a, int m) { // a**x = m mod p for (int i = 0, y = 1; i <= magic; ++i) { int inv = get<0>(ext_gcd(y, p)); if (inv < 0) inv += p; int u = 1LL * m * inv % p; if (mp.count(u)) return i * (p / magic) + mp[u]; y = 1LL * y * x % p; } return -1; } int generator() { // modulo prime p, O(x*log(phi(p))*log(p)), x = O(lg^6p) auto fp = PrimeFactor::factorize(p); int phi = p; for (auto u : fp) phi = phi / u.first * (u.first - 1); auto pp = PrimeFactor::factorize(phi); auto int_pow = [](int v, int n, int p) { int r = 1; for (int i = n; i; i >>= 1) { if (i & 1) r = 1LL * r * v % p; v = 1LL * v * v % p; } return r; }; auto check = [&](int g) { for (auto u : pp) { int t = int_pow(g, phi / u.first, p); if (t == 1) return false; } return true; }; for (int i = 2; ; ++i) { if (check(i)) return i; } return -1; } int discrete_kth_root(int n, int m) { // x**n = m mod p int q = discrete_log(g, m); if (q == -1) return -1; int x, y, d; tie(x, y, d) = ext_gcd(n, p - 1); if (q % d) return -1; // conclusion of no solution by ext_gcd int w = x; if (w < 0) w = w % ((p - 1) / d) + (p - 1) / d; // normalize to non-negative int r = 1; // g**q = g**(n*w') = (g**w')**n -> x = g**w' int ng = g; for (int64_t i = 1LL * q / d * w; i; i >>= 1) { if (i & 1) r = 1LL * r * ng % p; ng = 1LL * ng * ng % p; } return r; } int main() { ios::sync_with_stdio(false); int T; { cin >> T; } { x = 1; for (int i = 0; i < p / magic; ++i) { mp[x] = i; x = 1LL * x * g % p; } } while (T--) { int X, K; { cin >> X >> K; } cout << discrete_kth_root(K, X) << "\n"; } }