#include #define sz(v) ((int)(v).size()) #define all(v) (v).begin(), (v).end() using namespace std; typedef long long lint; typedef pair pi; const int MAXN = 30005; const int mod = 1e9 + 7; struct mint { int val; mint() { val = 0; } mint(const lint& v) { val = (-mod <= v && v < mod) ? v : v % mod; if (val < 0) val += mod; } friend ostream& operator<<(ostream& os, const mint& a) { return os << a.val; } friend bool operator==(const mint& a, const mint& b) { return a.val == b.val; } friend bool operator!=(const mint& a, const mint& b) { return !(a == b); } friend bool operator<(const mint& a, const mint& b) { return a.val < b.val; } mint operator-() const { return mint(-val); } mint& operator+=(const mint& m) { if ((val += m.val) >= mod) val -= mod; return *this; } mint& operator-=(const mint& m) { if ((val -= m.val) < 0) val += mod; return *this; } mint& operator*=(const mint& m) { val = (lint)val*m.val%mod; return *this; } friend mint ipow(mint a, lint p) { mint ans = 1; for (; p; p /= 2, a *= a) if (p&1) ans *= a; return ans; } friend mint inv(const mint& a) { assert(a.val); return ipow(a, mod - 2); } mint& operator/=(const mint& m) { return (*this) *= inv(m); } friend mint operator+(mint a, const mint& b) { return a += b; } friend mint operator-(mint a, const mint& b) { return a -= b; } friend mint operator*(mint a, const mint& b) { return a *= b; } friend mint operator/(mint a, const mint& b) { return a /= b; } operator int64_t() const {return val; } }; lint gcd(lint x, lint y){ return y ? gcd(y, x%y) : x; } lint mul(lint x, lint y, lint p){ return (__int128) x * y % p; } // find x such that x^k == a (mod p) // time: log^2 q + log q min(p^0.5, q^0.25) // https://judge.yosupo.jp/submission/78315 lint KthRootModPrime(lint a, lint k, lint p) { auto pow = [](lint a, lint n, lint p) -> lint { lint r = 1; for (; n > 0; n >>= 1, a = mul(a, a, p)) if (n % 2 == 1) r = mul(r, a, p); return r; }; auto inv = [](lint a, lint p) -> lint { a %= p; lint u = 1, v = 0; lint b = p; while (b > 0) { lint q = a / b; a %= b; u -= mul(v, q, p); u = (u % p + p) % p; swap(u, v); swap(a, b); } return u < 0 ? u + p : u; }; auto peth_root = [&](lint a, lint p, int e, lint mod) -> lint { lint q = mod - 1; int s = 0; while (q % p == 0) { q /= p; ++s; } lint pe = pow(p, e, mod); lint ans = pow(a, ((__int128)mul(pe - 1, inv(q, pe), pe) * q + 1) / pe, mod); lint c = 2; while (pow(c, (mod - 1) / p, mod) == 1) ++c; c = pow(c, q, mod); unordered_map map; lint add = 1; int v = (int)std::sqrt((double)(s - e) * p) + 1; lint mult = pow(c, mul(v, pow(p, s - 1, mod - 1), mod - 1), mod); for (int i = 0; i <= v; ++i) { map[add] = i; add = mul(add, mult, mod); } mult = inv(pow(c, pow(p, s - 1, mod - 1), mod), mod); for (int i = e; i < s; ++i) { lint err = mul(a, inv(pow(ans, pe, mod), mod), mod); lint target = pow(err, pow(p, s - 1 - i, mod - 1), mod); for (int j = 0; j <= v; ++j) { if (map.find(target) != map.end()) { int x = map[target]; ans = mul(ans, pow(c, mul(j + mul(v, x, mod - 1), pow(p, i - e, mod - 1), mod - 1), mod), mod); break; } target = mul(target, mult, mod); assert(j != v); } } return ans; }; if (k > 0 && a == 0) return 0; k %= p - 1; lint g = gcd(k, p - 1); if (pow(a, (p - 1) / g, p) != 1) return -1; a = pow(a, inv(k / g, (p - 1) / g), p); for (lint div = 2; div * div <= g; ++div) { int sz = 0; while (g % div == 0) { g /= div; ++sz; } if (sz > 0) { lint b = peth_root(a, div, sz, p); a = b; } } if (g > 1) a = peth_root(a, g, 1, p); return a; } int main(){ ios::sync_with_stdio(false); cin.tie(0); cout.tie(0); int tc; cin >> tc; while(tc--){ lint p, k, a; cin >> p >> k >> a; cout << KthRootModPrime(a, k, p) << "\n"; } /* int n; cin >> n; vector> v; for(int i = 0; i < n; i++){ string s; cin >> s; mint ans = 0; for(auto &i : s){ ans = ans * mint(10) + mint(i - '0'); } mint x = ipow(ans, mod / 2); mint y = ipow(-ans, mod / 2); cout << x*x << " " << y*y << "\n"; if(x * x != ans){ assert(y * y == -ans); v.emplace_back(0, y); } else v.emplace_back(x, 0); } */ }