結果

問題 No.981 一般冪乗根
ユーザー maspymaspy
提出日時 2022-04-25 12:57:43
言語 C++17
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 12 ms / 6,000 ms
コード長 19,795 bytes
コンパイル時間 2,961 ms
コンパイル使用メモリ 228,356 KB
実行使用メモリ 5,376 KB
最終ジャッジ日時 2024-06-27 08:22:50
合計ジャッジ時間 47,321 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge3
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 6 ms
5,248 KB
testcase_01 AC 6 ms
5,376 KB
testcase_02 AC 6 ms
5,376 KB
testcase_03 AC 6 ms
5,376 KB
testcase_04 AC 6 ms
5,376 KB
testcase_05 AC 5 ms
5,376 KB
testcase_06 AC 5 ms
5,376 KB
testcase_07 AC 5 ms
5,376 KB
testcase_08 AC 4 ms
5,376 KB
testcase_09 AC 5 ms
5,376 KB
testcase_10 AC 5 ms
5,376 KB
testcase_11 AC 5 ms
5,376 KB
testcase_12 AC 5 ms
5,376 KB
testcase_13 AC 5 ms
5,376 KB
testcase_14 AC 5 ms
5,376 KB
testcase_15 AC 5 ms
5,376 KB
testcase_16 AC 5 ms
5,376 KB
testcase_17 AC 4 ms
5,376 KB
testcase_18 AC 4 ms
5,376 KB
testcase_19 AC 5 ms
5,376 KB
testcase_20 AC 5 ms
5,376 KB
testcase_21 AC 4 ms
5,376 KB
testcase_22 AC 5 ms
5,376 KB
testcase_23 AC 5 ms
5,376 KB
testcase_24 AC 4 ms
5,376 KB
testcase_25 AC 6 ms
5,376 KB
testcase_26 AC 6 ms
5,376 KB
testcase_27 AC 3 ms
5,376 KB
testcase_28 AC 12 ms
5,376 KB
evil_60bit1.txt RE -
evil_60bit2.txt RE -
evil_60bit3.txt RE -
evil_hack AC 2 ms
5,376 KB
evil_hard_random RE -
evil_hard_safeprime.txt RE -
evil_hard_tonelli0 RE -
evil_hard_tonelli1 RE -
evil_hard_tonelli2 RE -
evil_hard_tonelli3 RE -
evil_sefeprime1.txt RE -
evil_sefeprime2.txt RE -
evil_sefeprime3.txt RE -
evil_tonelli1.txt RE -
evil_tonelli2.txt RE -
権限があれば一括ダウンロードができます

ソースコード

diff # 

#line 1 "/home/maspy/compro/library/my_template.hpp"
#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using pi = pair<ll, ll>;
using vi = vector<ll>;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;

template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vec(type, name, ...) vector<type> name(__VA_ARGS__)
#define vv(type, name, h, ...) \
  vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...)   \
  vector<vector<vector<type>>> name( \
      h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...)       \
  vector<vector<vector<vector<type>>>> name( \
      a, vector<vector<vector<type>>>(       \
             b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

#define FOR_(n) for (ll _ = 0; (_) < (ll)(n); ++(_))
#define FOR(i, n) for (ll i = 0; (i) < (ll)(n); ++(i))
#define FOR3(i, m, n) for (ll i = (m); (i) < (ll)(n); ++(i))
#define FOR_R(i, n) for (ll i = (ll)(n)-1; (i) >= 0; --(i))
#define FOR3_R(i, m, n) for (ll i = (ll)(n)-1; (i) >= (ll)(m); --(i))
#define FOR_subset(t, s) for (ll t = s; t >= 0; t = (t == 0 ? -1 : (t - 1) & s))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

template <typename T>
T SUM(vector<T> &A) {
  T sum = T(0);
  for (auto &&a: A) sum += a;
  return sum;
}
#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end())

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }

template <typename T, typename U>
T ceil(T x, U y) {
  return (x > 0 ? (x + y - 1) / y : x / y);
}

template <typename T, typename U>
T floor(T x, U y) {
  return (x > 0 ? x / y : (x - y + 1) / y);
}

template <typename T, typename U>
pair<T, T> divmod(T x, U y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

ll binary_search(function<bool(ll)> check, ll ok, ll ng) {
  assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    if (check(x))
      ok = x;
    else
      ng = x;
  }
  return ok;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

vi s_to_vi(const string& S, char first_char) {
  vi A(S.size());
  FOR(i, S.size()) { A[i] = S[i] - first_char; }
  return A;
}

template <typename T>
vector<T> cumsum(vector<T> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

template <typename CNT, typename T>
vc<CNT> bincount(const vc<T> &A, int size) {
  vc<CNT> C(size);
  for (auto &&x: A) { ++C[x]; }
  return C;
}

template <typename T>
vector<int> argsort(const vector<T> &A) {
  // stable
  vector<int> ids(A.size());
  iota(all(ids), 0);
  sort(all(ids),
       [&](int i, int j) { return A[i] < A[j] || (A[i] == A[j] && i < j); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  int n = len(A);
  assert(len(I) == n);
  vc<T> B(n);
  FOR(i, n) B[i] = A[I[i]];
  return B;
}
#line 1 "/home/maspy/compro/library/other/io.hpp"
// based on yosupo's fastio
#include <unistd.h>

namespace detail {
template <typename T, decltype(&T::is_modint) = &T::is_modint>
std::true_type check_value(int);
template <typename T>
std::false_type check_value(long);
} // namespace detail

template <typename T>
struct is_modint : decltype(detail::check_value<T>(0)) {};
template <typename T>
using is_modint_t = enable_if_t<is_modint<T>::value>;
template <typename T>
using is_not_modint_t = enable_if_t<!is_modint<T>::value>;

struct Scanner {
  FILE *fp;
  char line[(1 << 15) + 1];
  size_t st = 0, ed = 0;
  void reread() {
    memmove(line, line + st, ed - st);
    ed -= st;
    st = 0;
    ed += fread(line + ed, 1, (1 << 15) - ed, fp);
    line[ed] = '\0';
  }
  bool succ() {
    while (true) {
      if (st == ed) {
        reread();
        if (st == ed) return false;
      }
      while (st != ed && isspace(line[st])) st++;
      if (st != ed) break;
    }
    if (ed - st <= 50) {
      bool sep = false;
      for (size_t i = st; i < ed; i++) {
        if (isspace(line[i])) {
          sep = true;
          break;
        }
      }
      if (!sep) reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_same<T, string>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    while (true) {
      size_t sz = 0;
      while (st + sz < ed && !isspace(line[st + sz])) sz++;
      ref.append(line + st, sz);
      st += sz;
      if (!sz || st != ed) break;
      reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    bool neg = false;
    if (line[st] == '-') {
      neg = true;
      st++;
    }
    ref = T(0);
    while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); }
    if (neg) ref = -ref;
    return true;
  }
  template <class T, is_modint_t<T> * = nullptr>
  bool read_single(T &ref) {
    long long val = 0;
    bool f = read_single(val);
    ref = T(val);
    return f;
  }
  bool read_single(double &ref) {
    string s;
    if (!read_single(s)) return false;
    ref = std::stod(s);
    return true;
  }
  bool read_single(char &ref) {
    string s;
    if (!read_single(s) || s.size() != 1) return false;
    ref = s[0];
    return true;
  }
  template <class T>
  bool read_single(vector<T> &ref) {
    for (auto &d: ref) {
      if (!read_single(d)) return false;
    }
    return true;
  }
  template <class T, class U>
  bool read_single(pair<T, U> &p) {
    return (read_single(p.first) && read_single(p.second));
  }
  template <class A, class B, class C>
  bool read_single(tuple<A, B, C> &p) {
    return (read_single(get<0>(p)) && read_single(get<1>(p))
            && read_single(get<2>(p)));
  }
  template <class A, class B, class C, class D>
  bool read_single(tuple<A, B, C, D> &p) {
    return (read_single(get<0>(p)) && read_single(get<1>(p))
            && read_single(get<2>(p)) && read_single(get<3>(p)));
  }
  void read() {}
  template <class H, class... T>
  void read(H &h, T &... t) {
    bool f = read_single(h);
    assert(f);
    read(t...);
  }
  Scanner(FILE *fp) : fp(fp) {}
};

struct Printer {
  Printer(FILE *_fp) : fp(_fp) {}
  ~Printer() { flush(); }

  static constexpr size_t SIZE = 1 << 15;
  FILE *fp;
  char line[SIZE], small[50];
  size_t pos = 0;
  void flush() {
    fwrite(line, 1, pos, fp);
    pos = 0;
  }
  void write(const char &val) {
    if (pos == SIZE) flush();
    line[pos++] = val;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  void write(T val) {
    if (pos > (1 << 15) - 50) flush();
    if (val == 0) {
      write('0');
      return;
    }
    if (val < 0) {
      write('-');
      val = -val; // todo min
    }
    size_t len = 0;
    while (val) {
      small[len++] = char(0x30 | (val % 10));
      val /= 10;
    }
    for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; }
    pos += len;
  }
  void write(const string &s) {
    for (char c: s) write(c);
  }
  void write(const char *s) {
    size_t len = strlen(s);
    for (size_t i = 0; i < len; i++) write(s[i]);
  }
  void write(const double &x) {
    ostringstream oss;
    oss << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  void write(const long double &x) {
    ostringstream oss;
    oss << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  template <class T, is_modint_t<T> * = nullptr>
  void write(T &ref) {
    write(ref.val);
  }
  template <class T>
  void write(const vector<T> &val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  template <class T, class U>
  void write(const pair<T, U> &val) {
    write(val.first);
    write(' ');
    write(val.second);
  }
  template <class A, class B, class C>
  void write(const tuple<A, B, C> &val) {
    auto &[a, b, c] = val;
    write(a), write(' '), write(b), write(' '), write(c);
  }
  template <class A, class B, class C, class D>
  void write(const tuple<A, B, C, D> &val) {
    auto &[a, b, c, d] = val;
    write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d);
  }
  template <class A, class B, class C, class D, class E>
  void write(const tuple<A, B, C, D, E> &val) {
    auto &[a, b, c, d, e] = val;
    write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d), write(' '), write(e);
  }
  template <class A, class B, class C, class D, class E, class F>
  void write(const tuple<A, B, C, D, E, F> &val) {
    auto &[a, b, c, d, e, f] = val;
    write(a), write(' '), write(b), write(' '), write(c), write(' '), write(d), write(' '), write(e), write(' '), write(f);
  }
  template <class T, size_t S>
  void write(const array<T, S> &val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  void write(i128 val) {
    string s;
    bool negative = 0;
    if(val < 0){
      negative = 1;
      val = -val;
    }
    while (val) {
      s += '0' + int(val % 10);
      val /= 10;
    }
    if(negative) s += "-";
    reverse(all(s));
    if (len(s) == 0) s = "0";
    write(s);
  }
};

Scanner scanner = Scanner(stdin);
Printer printer = Printer(stdout);

void flush() { printer.flush(); }
void print() { printer.write('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  printer.write(head);
  if (sizeof...(Tail)) printer.write(' ');
  print(forward<Tail>(tail)...);
}

void read() {}
template <class Head, class... Tail>
void read(Head &head, Tail &... tail) {
  scanner.read(head);
  read(tail...);
}

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __VA_ARGS__; \
  read(__VA_ARGS__)
#define STR(...)      \
  string __VA_ARGS__; \
  read(__VA_ARGS__)
#define CHAR(...)      \
  char __VA_ARGS__; \
  read(__VA_ARGS__)
#define DBL(...)      \
  double __VA_ARGS__; \
  read(__VA_ARGS__)

#define VEC(type, name, size) \
  vector<type> name(size);    \
  read(name)
#define VV(type, name, h, w)                     \
  vector<vector<type>> name(h, vector<type>(w)); \
  read(name)

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 2 "/home/maspy/compro/library/nt/primetest.hpp"
struct m64 {
    using i64 = int64_t;
    using u64 = uint64_t;
    using u128 = __uint128_t;

    inline static u64 m, r, n2; // r * m = -1 (mod 1<<64), n2 = 1<<128 (mod m)
    static void set_mod(u64 m) {
        assert(m < (1ull << 62));
        assert((m & 1) == 1);
        m64::m = m;
        n2 = -u128(m) % m;
        r = m;
        FOR (_, 5) r *= 2 - m*r;
        r = -r;
        assert(r * m == -1ull);
    }
    static u64 reduce(u128 b) { return (b + u128(u64(b) * r) * m) >> 64; }

    u64 x;
    m64() : x(0) {}
    m64(u64 x) : x(reduce(u128(x) * n2)){};
    u64 val() const { u64 y = reduce(x); return y >= m ? y-m : y; }
    m64 &operator+=(m64 y) {
        x += y.x - (m << 1);
        x = (i64(x) < 0 ? x + (m << 1) : x);
        return *this;
    }
    m64 &operator-=(m64 y) {
        x -= y.x;
        x = (i64(x) < 0 ? x + (m << 1) : x);
        return *this;
    }
    m64 &operator*=(m64 y) { x = reduce(u128(x) * y.x); return *this; }
    m64 operator+(m64 y) const { return m64(*this) += y; }
    m64 operator-(m64 y) const { return m64(*this) -= y; }
    m64 operator*(m64 y) const { return m64(*this) *= y; }
    bool operator==(m64 y) const { return (x >= m ? x-m : x) == (y.x >= m ? y.x-m : y.x); }
    bool operator!=(m64 y) const { return not operator==(y); }
    m64 pow(u64 n) const {
        m64 y = 1, z = *this;
        for ( ; n; n >>= 1, z *= z) if (n & 1) y *= z;
        return y;
    }
};

bool primetest(const uint64_t x) {
    using u64 = uint64_t;
    if (x == 2 or x == 3 or x == 5 or x == 7) return true;
    if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return false;
    if (x < 121) return x > 1;
    const u64 d = (x-1) >> __builtin_ctzll(x-1);
    m64::set_mod(x);
    const m64 one(1), minus_one(x-1);
    auto ok = [&](u64 a) {
        auto y = m64(a).pow(d);
        u64 t = d;
        while (y != one and y != minus_one and t != x-1) y *= y, t <<= 1;
        if (y != minus_one and t % 2 == 0) return false;
        return true;
    };
    if (x < (1ull << 32)) {
        for (u64 a : { 2, 7, 61 }) if (not ok(a)) return false;
    } else {
        for (u64 a : { 2, 325, 9375, 28178, 450775, 9780504, 1795265022 }) {
            if (x <= a) return true;
            if (not ok(a)) return false;
        }
    }
    return true;
}
#line 3 "/home/maspy/compro/library/nt/factor.hpp"

mt19937_64 rng_mt{random_device{}()};
ll rnd(ll n) { return uniform_int_distribution<ll>(0, n - 1)(rng_mt); }

ll rho(ll n, ll c) {
  m64::set_mod(n);
  assert(n > 1);
  const m64 cc(c);
  auto f = [&](m64 x) { return x * x + cc; };
  m64 x = 1, y = 2, z = 1, q = 1;
  ll g = 1;
  const ll m = 1LL << (__lg(n) / 5); // ?
  for (ll r = 1; g == 1; r <<= 1) {
    x = y;
    FOR(_, r) y = f(y);
    for (ll k = 0; k < r and g == 1; k += m) {
      z = y;
      FOR(_, min(m, r - k)) y = f(y), q *= x - y;
      g = gcd(q.val(), n);
    }
  }
  if (g == n)
    do {
      z = f(z);
      g = gcd((x - z).val(), n);
    } while (g == 1);
  return g;
}

ll find_prime_factor(ll n) {
  assert(n > 1);
  if (primetest(n))
    return n;
  FOR(_, 100) {
    ll m = rho(n, rnd(n));
    if (primetest(m))
      return m;
    n = m;
  }
  cerr << "failed" << endl;
  assert(false);
  return -1;
}

vc<pi> factor(ll n) {
  assert(n >= 1);
  vc<pi> pf;
  FOR3(p, 2, 100) {
    if (p * p > n)
      break;
    if (n % p == 0) {
      ll e = 0;
      do {
        n /= p, e += 1;
      } while (n % p == 0);
      pf.eb(p, e);
    }
  }
  while (n > 1) {
    ll p = find_prime_factor(n);
    ll e = 0;
    do {
      n /= p, e += 1;
    } while (n % p == 0);
    pf.eb(p, e);
  }
  sort(all(pf));
  return pf;
}
#line 1 "/home/maspy/compro/library/mod/fast_div.hpp"
struct fast_div {
  // Min25 https://judge.yosupo.jp/submission/46090
  // 同じ定数で何度も除算するときの高速化に使える
  using i64 = long long;
  using u64 = unsigned long long;
  using u128 = __uint128_t;
  constexpr fast_div() : m(), s(), x() {}
  constexpr fast_div(int n)
      : m(n), s(std::__lg(n - 1)), x(((u128(1) << (s + 64)) + n - 1) / n) {}
  constexpr friend u64 operator/(u64 n, const fast_div& d) {
    return (u128(n) * d.x >> d.s) >> 64;
  }
  constexpr friend int operator%(u64 n, const fast_div& d) {
    return n - n / d * d.m;
  }
  constexpr std::pair<i64, int> divmod(u64 n) const {
    u64 q = n / *this;
    return {q, n - q * m};
  }

  int m;
  int s;
  u64 x;
};
#line 2 "/home/maspy/compro/library/mod/mod_pow.hpp"
ll mod_pow(ll a, ll n, int mod){
  fast_div fd(mod);
  a = a % fd;
  ll p = a;
  ll v = 1;
  while(n){
    if(n & 1) v = v * p % fd;
    p = p * p % fd;
    n >>= 1;
  }
  return v;
}
#line 3 "/home/maspy/compro/library/mod/primitive_root.hpp"
int primitive_root(int p) {
  auto pf = factor(p - 1);
  auto is_ok = [&](int g) -> bool {
    for (auto&& [q, e]: pf)
      if (mod_pow(g, (p - 1) / q, p) == 1) return false;
    return true;
  };
  FOR3(x, 1, p) {
    if (is_ok(x)) return x;
  }
  return -1;
}
#line 1 "/home/maspy/compro/library/mod/mod_inv.hpp"
ll mod_inv(ll val, ll mod) {
  int a = val, b = mod, u = 1, v = 0, t;
  while (b > 0) {
    t = a / b;
    swap(a -= t * b, b), swap(u -= t * v, v);
  }
  if(u < 0) u += mod;
  return u;
}
#line 4 "/home/maspy/compro/library/mod/mod_kth_root.hpp"

int mod_kth_root(ll k, ll a, int mod) {
  assert(primetest(mod) && 0 <= a && a < mod);
  if (k == 0) return (a == 1 ? 1 : -1);
  if (a == 0) return 0;
  if (mod == 2) return a;
  k %= mod - 1;
  fast_div fd(mod);

  ll g = gcd(k, mod - 1);
  if (mod_pow(a, (mod - 1) / g, mod) != 1) return -1;

  ll c = mod_inv(k / g, (mod - 1) / g);
  a = mod_pow(a, c, mod);
  k = (k * c) % (mod - 1);
  if (k == 0) return 1;

  g = primitive_root(mod);

  auto solve_pp = [&](ll p, int e, ll a) -> ll {
    int f = 0;
    ll pf = 1;
    while ((mod - 1) % (pf * p) == 0) ++f, pf *= p;
    ll m = (mod - 1) / pf;
    /*
    ・位数 Qm の巡回群
    ・a の p^e 乗根をとりたい。持つことは分かっている
    ・a / x^{p^e} = b を維持する。まずは、b が p で割れる回数を増やしていく。
    */
    ll x = 1, b = a, c = f - e; // b ^ {mp^c} = 1
    int pc = 1;
    FOR_(c) pc *= p;
    int pe = 1;
    FOR_(e) pe *= p;
    // 必要ならば原始 p 乗根に関する離散対数問題のセットアップ
    ll G = mod_pow(g, (mod - 1) / p, mod);
    int M = 0;
    unordered_map<ll, int> MP;
    ll GM_inv = -1;
    if (c) {
      while (M * M < p) ++M;
      MP.reserve(M + 1);
      ll Gpow = 1;
      FOR(m, M) {
        MP[Gpow] = m;
        Gpow = Gpow * G % fd;
      }
      GM_inv = mod_pow(Gpow, mod - 2, mod);
    }

    while (c) {
      /*
      b^{mp^c} = 1 が分かっている。(b/x^{p^e}})^{mp^{c-1}} = 1 にしたい。
      x = g^{p^{f-c-e}*k} として探す。原始 p 乗根 B, G に対する B = G^k に帰着。
      */
      ll B = mod_pow(b, m * pc / p, mod);
      int k = [&](ll B) -> int {
        FOR(m, M + 1) {
          if (MP.count(B)) return m * M + MP[B];
          B = B * GM_inv % fd;
        }
        return -1;
      }(B);
      x = x * mod_pow(g, pf / pc / pe * k, mod) % fd;
      ll exp = pf / pc * k % (mod - 1);
      b = b * mod_pow(g, mod - 1 - exp, mod) % fd;
      --c;
      pc /= p;
    }
    int k = pe - mod_inv(m, pe);
    k = (k * m + 1) / pe;
    ll y = mod_pow(b, k, mod);
    x = x * y % fd;
    return x;
  };

  auto pf = factor(k);
  for (auto&& [p, e]: pf) a = solve_pp(p, e, a);
  return a;
}
#line 4 "main.cpp"

void solve() {
  LL(p, k, a);
  ll ANS = mod_kth_root(k, a, p);
  print(ANS);
}

signed main() {
  cin.tie(nullptr);
  ios::sync_with_stdio(false);
  cout << setprecision(15);

  LL(T);
  FOR(_, T) solve();

  return 0;
}
0