#line 1 "sol.cpp" #line 1 "library/Template/template.hpp" #include using namespace std; #define rep(i, a, b) for (int i = (int)(a); i < (int)(b); i++) #define rrep(i, a, b) for (int i = (int)(b)-1; i >= (int)(a); i--) #define ALL(v) (v).begin(), (v).end() #define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end()) #define SZ(v) (int)v.size() #define MIN(v) *min_element(ALL(v)) #define MAX(v) *max_element(ALL(v)) #define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin()) #define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin()) using uint = unsigned int; using ll = long long int; using ull = unsigned long long; using i128 = __int128_t; using u128 = __uint128_t; const int inf = 0x3fffffff; const ll INF = 0x1fffffffffffffff; template inline bool chmax(T &a, T b) { if (a < b) { a = b; return 1; } return 0; } template inline bool chmin(T &a, T b) { if (a > b) { a = b; return 1; } return 0; } template T ceil(T x, U y) { assert(y != 0); if (y < 0) x = -x, y = -y; return (x > 0 ? (x + y - 1) / y : x / y); } template T floor(T x, U y) { assert(y != 0); if (y < 0) x = -x, y = -y; return (x > 0 ? x / y : (x - y + 1) / y); } template int popcnt(T x) { return __builtin_popcountll(x); } template int topbit(T x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } template int lowbit(T x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template ostream &operator<<(ostream &os, const pair &p) { os << "P(" << p.first << ", " << p.second << ")"; return os; } template ostream &operator<<(ostream &os, const vector &vec) { os << "{"; for (int i = 0; i < vec.size(); i++) { os << vec[i] << (i + 1 == vec.size() ? "" : ", "); } os << "}"; return os; } template ostream &operator<<(ostream &os, const map &map_var) { os << "{"; for (auto itr = map_var.begin(); itr != map_var.end(); itr++) { os << "(" << itr->first << ", " << itr->second << ")"; itr++; if (itr != map_var.end()) os << ", "; itr--; } os << "}"; return os; } template ostream &operator<<(ostream &os, const set &set_var) { os << "{"; for (auto itr = set_var.begin(); itr != set_var.end(); itr++) { os << *itr; ++itr; if (itr != set_var.end()) os << ", "; itr--; } os << "}"; return os; } #ifdef LOCAL #define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__) #else #define show(...) true #endif template void _show(int i, T name) { cerr << '\n'; } template void _show(int i, const T1 &a, const T2 &b, const T3 &...c) { for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i]; cerr << ":" << b << " "; _show(i + 1, a, c...); } #line 2 "library/Utility/fastio.hpp" #include namespace fastio { static constexpr uint32_t SZ = 1 << 17; char ibuf[SZ]; char obuf[SZ]; char out[100]; // pointer of ibuf, obuf uint32_t pil = 0, pir = 0, por = 0; struct Pre { char num[10000][4]; constexpr Pre() : num() { for (int i = 0; i < 10000; i++) { int n = i; for (int j = 3; j >= 0; j--) { num[i][j] = n % 10 | '0'; n /= 10; } } } } constexpr pre; inline void load() { memmove(ibuf, ibuf + pil, pir - pil); pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin); pil = 0; if (pir < SZ) ibuf[pir++] = '\n'; } inline void flush() { fwrite(obuf, 1, por, stdout); por = 0; } void rd(char &c) { do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); } void rd(string &x) { x.clear(); char c; do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); do { x += c; if (pil == pir) load(); c = ibuf[pil++]; } while (!isspace(c)); } template void rd_real(T &x) { string s; rd(s); x = stod(s); } template void rd_integer(T &x) { if (pil + 100 > pir) load(); char c; do c = ibuf[pil++]; while (c < '-'); bool minus = 0; if constexpr (is_signed::value || is_same_v) { if (c == '-') { minus = 1, c = ibuf[pil++]; } } x = 0; while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; } if constexpr (is_signed::value || is_same_v) { if (minus) x = -x; } } void rd(int &x) { rd_integer(x); } void rd(ll &x) { rd_integer(x); } void rd(i128 &x) { rd_integer(x); } void rd(uint &x) { rd_integer(x); } void rd(ull &x) { rd_integer(x); } void rd(u128 &x) { rd_integer(x); } void rd(double &x) { rd_real(x); } void rd(long double &x) { rd_real(x); } template void rd(pair &p) { return rd(p.first), rd(p.second); } template void rd_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); rd(x); rd_tuple(t); } } template void rd(tuple &tpl) { rd_tuple(tpl); } template void rd(array &x) { for (auto &d : x) rd(d); } template void rd(vector &x) { for (auto &d : x) rd(d); } void read() {} template void read(H &h, T &...t) { rd(h), read(t...); } void wt(const char c) { if (por == SZ) flush(); obuf[por++] = c; } void wt(const string s) { for (char c : s) wt(c); } void wt(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) wt(s[i]); } template void wt_integer(T x) { if (por > SZ - 100) flush(); if (x < 0) { obuf[por++] = '-', x = -x; } int outi; for (outi = 96; x >= 10000; outi -= 4) { memcpy(out + outi, pre.num[x % 10000], 4); x /= 10000; } if (x >= 1000) { memcpy(obuf + por, pre.num[x], 4); por += 4; } else if (x >= 100) { memcpy(obuf + por, pre.num[x] + 1, 3); por += 3; } else if (x >= 10) { int q = (x * 103) >> 10; obuf[por] = q | '0'; obuf[por + 1] = (x - q * 10) | '0'; por += 2; } else obuf[por++] = x | '0'; memcpy(obuf + por, out + outi + 4, 96 - outi); por += 96 - outi; } template void wt_real(T x) { ostringstream oss; oss << fixed << setprecision(15) << double(x); string s = oss.str(); wt(s); } void wt(int x) { wt_integer(x); } void wt(ll x) { wt_integer(x); } void wt(i128 x) { wt_integer(x); } void wt(uint x) { wt_integer(x); } void wt(ull x) { wt_integer(x); } void wt(u128 x) { wt_integer(x); } void wt(double x) { wt_real(x); } void wt(long double x) { wt_real(x); } template void wt(const pair val) { wt(val.first); wt(' '); wt(val.second); } template void wt_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { wt(' '); } const auto x = std::get(t); wt(x); wt_tuple(t); } } template void wt(tuple tpl) { wt_tuple(tpl); } template void wt(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } template void wt(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } void print() { wt('\n'); } template void print(Head &&head, Tail &&...tail) { wt(head); if (sizeof...(Tail)) wt(' '); print(forward(tail)...); } void __attribute__((destructor)) _d() { flush(); } } // namespace fastio using fastio::flush; using fastio::print; using fastio::read; inline void first(bool i = true) { print(i ? "first" : "second"); } inline void Alice(bool i = true) { print(i ? "Alice" : "Bob"); } inline void Takahashi(bool i = true) { print(i ? "Takahashi" : "Aoki"); } inline void yes(bool i = true) { print(i ? "yes" : "no"); } inline void Yes(bool i = true) { print(i ? "Yes" : "No"); } inline void No() { print("No"); } inline void YES(bool i = true) { print(i ? "YES" : "NO"); } inline void NO() { print("NO"); } inline void Yay(bool i = true) { print(i ? "Yay!" : ":("); } inline void Possible(bool i = true) { print(i ? "Possible" : "Impossible"); } inline void POSSIBLE(bool i = true) { print(i ? "POSSIBLE" : "IMPOSSIBLE"); } /** * @brief Fast IO */ #line 3 "sol.cpp" #line 2 "library/Math/fastdiv.hpp" struct FastDiv { using u64 = uint64_t; using u128 = __uint128_t; constexpr FastDiv() : m(), s(), x() {} constexpr FastDiv(ll _m) : m(_m), s(__lg(m - 1)), x(((u128(1) << (s + 64)) + m - 1) / m) {} constexpr ll get() { return m; } constexpr friend u64 operator/(u64 n, const FastDiv &d) { return (u128(n) * d.x >> d.s) >> 64; } constexpr friend ll operator%(u64 n, const FastDiv &d) { return n - n / d * d.m; } constexpr pair divmod(u64 n) const { u64 q = n / (*this); return {q, n - q * m}; } ll m, s; u64 x; }; /** * @brief Fast Division */ #line 2 "library/Math/miller.hpp" struct m64 { using i64 = int64_t; using u64 = uint64_t; using u128 = __uint128_t; static u64 mod; static u64 r; static u64 n2; static u64 get_r() { u64 ret = mod; rep(_, 0, 5) ret *= 2 - mod * ret; return ret; } static void set_mod(u64 m) { assert(m < (1LL << 62)); assert((m & 1) == 1); mod = m; n2 = -u128(m) % m; r = get_r(); assert(r * mod == 1); } static u64 get_mod() { return mod; } u64 a; m64() : a(0) {} m64(const int64_t &b) : a(reduce((u128(b) + mod) * n2)){}; static u64 reduce(const u128 &b) { return (b + u128(u64(b) * u64(-r)) * mod) >> 64; } u64 get() const { u64 ret = reduce(a); return ret >= mod ? ret - mod : ret; } m64 &operator*=(const m64 &b) { a = reduce(u128(a) * b.a); return *this; } m64 operator*(const m64 &b) const { return m64(*this) *= b; } bool operator==(const m64 &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } bool operator!=(const m64 &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } m64 pow(u128 n) const { m64 ret(1), mul(*this); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } }; typename m64::u64 m64::mod, m64::r, m64::n2; bool Miller(ll n) { if (n < 2 or (n & 1) == 0) return (n == 2); m64::set_mod(n); ll d = n - 1; while ((d & 1) == 0) d >>= 1; vector seeds; if (n < (1 << 30)) seeds = {2, 7, 61}; else seeds = {2, 325, 9375, 28178, 450775, 9780504}; for (auto &x : seeds) { if (n <= x) break; ll t = d; m64 y = m64(x).pow(t); while (t != n - 1 and y != 1 and y != n - 1) { y *= y; t <<= 1; } if (y != n - 1 and (t & 1) == 0) return 0; } return 1; } /** * @brief Miller-Rabin */ #line 2 "library/Utility/random.hpp" namespace Random { mt19937_64 randgen(chrono::steady_clock::now().time_since_epoch().count()); using u64 = unsigned long long; u64 get() { return randgen(); } template T get(T L) { // [0,L] return get() % (L + 1); } template T get(T L, T R) { // [L,R] return get(R - L) + L; } double uniform() { return double(get(1000000000)) / 1000000000; } string str(int n) { string ret; rep(i, 0, n) ret += get('a', 'z'); return ret; } template void shuffle(Iter first, Iter last) { if (first == last) return; int len = 1; for (auto it = first + 1; it != last; it++) { len++; int j = get(0, len - 1); if (j != len - 1) iter_swap(it, first + j); } } template vector select(int n, T L, T R) { // [L,R] if (n * 2 >= R - L + 1) { vector ret(R - L + 1); iota(ALL(ret), L); shuffle(ALL(ret)); ret.resize(n); return ret; } else { unordered_set used; vector ret; while (SZ(used) < n) { T x = get(L, R); if (!used.count(x)) { used.insert(x); ret.push_back(x); } } return ret; } } void relabel(int n, vector> &es) { shuffle(ALL(es)); vector ord(n); iota(ALL(ord), 0); shuffle(ALL(ord)); for (auto &[u, v] : es) u = ord[u], v = ord[v]; } template vector> genGraph(int n) { vector> cand, es; rep(u, 0, n) rep(v, 0, n) { if (simple and u == v) continue; if (!directed and u > v) continue; cand.push_back({u, v}); } int m = get(SZ(cand)); vector ord; if (simple) ord = select(m, 0, SZ(cand) - 1); else { rep(_, 0, m) ord.push_back(get(SZ(cand) - 1)); } for (auto &i : ord) es.push_back(cand[i]); relabel(n, es); return es; } vector> genTree(int n) { vector> es; rep(i, 1, n) es.push_back({get(i - 1), i}); relabel(n, es); return es; } }; // namespace Random /** * @brief Random */ #line 4 "library/Math/pollard.hpp" vector Pollard(ll n) { if (n <= 1) return {}; if (Miller(n)) return {n}; if ((n & 1) == 0) { vector v = Pollard(n >> 1); v.push_back(2); return v; } for (ll x = 2, y = 2, d;;) { ll c = Random::get(2LL, n - 1); do { x = (__int128_t(x) * x + c) % n; y = (__int128_t(y) * y + c) % n; y = (__int128_t(y) * y + c) % n; d = __gcd(x - y + n, n); } while (d == 1); if (d < n) { vector lb = Pollard(d), rb = Pollard(n / d); lb.insert(lb.end(), ALL(rb)); return lb; } } } /** * @brief Pollard-Rho */ #line 4 "library/Math/primitive.hpp" ll mpow(ll a, ll t, ll m) { ll res = 1; FastDiv im(m); while (t) { if (t & 1) res = __int128_t(res) * a % m; a = __int128_t(a) * a % m; t >>= 1; } return res; } ll minv(ll a, ll m) { ll b = m, u = 1, v = 0; while (b) { ll t = a / b; a -= t * b; swap(a, b); u -= t * v; swap(u, v); } u = (u % m + m) % m; return u; } ll getPrimitiveRoot(ll p) { vector ps = Pollard(p - 1); sort(ALL(ps)); rep(x, 1, inf) { for (auto &q : ps) { if (mpow(x, (p - 1) / q, p) == 1) goto fail; } return x; fail:; } assert(0); } ll extgcd(ll a, ll b, ll &p, ll &q) { if (b == 0) { p = 1; q = 0; return a; } ll d = extgcd(b, a % b, q, p); q -= a / b * p; return d; } pair crt(const vector &vs, const vector &ms) { ll V = vs[0], M = ms[0]; rep(i, 1, vs.size()) { ll p, q, v = vs[i], m = ms[i]; if (M < m) swap(M, m), swap(V, v); ll d = extgcd(M, m, p, q); if ((v - V) % d != 0) return {0, -1}; ll md = m / d, tmp = (v - V) / d % md * p % md; V += M * tmp; M *= md; } V = (V % M + M) % M; return {V, M}; } ll ModLog(ll a, ll b, ll p) { ll g = 1; for (ll t = p; t; t >>= 1) g = g * a % p; g = __gcd(g, p); ll t = 1, c = 0; for (; t % g; c++) { if (t == b) return c; t = t * a % p; } if (b % g) return -1; t /= g, b /= g; ll n = p / g, h = 0, gs = 1; for (; h * h < n; h++) gs = gs * a % n; unordered_map bs; for (ll s = 0, e = b; s < h; bs[e] = ++s) e = e * a % n; for (ll s = 0, e = t; s < n;) { e = e * gs % n, s += h; if (bs.count(e)) { return c + s - bs[e]; } } return -1; } ll mod_root(ll k, ll a, ll m) { if (a == 0) return k ? 0 : -1; if (m == 2) return a & 1; k %= m - 1; ll g = gcd(k, m - 1); if (mpow(a, (m - 1) / g, m) != 1) return -1; a = mpow(a, minv(k / g, (m - 1) / g), m); FastDiv im(m); auto _subroot = [&](ll p, int e, ll a) -> ll { // x^(p^e)==a(mod m) ll q = m - 1; int s = 0; while (q % p == 0) { q /= p; s++; } int d = s - e; ll pe = mpow(p, e, m), res = mpow(a, ((pe - 1) * minv(q, pe) % pe * q + 1) / pe, m), c = 1; while (mpow(c, (m - 1) / p, m) == 1) c++; c = mpow(c, q, m); map mp; ll v = 1, block = sqrt(d * p) + 1, bs = mpow(c, mpow(p, s - 1, m - 1) * block % (m - 1), m); rep(i, 0, block + 1) mp[v] = i, v = v * bs % im; ll gs = minv(mpow(c, mpow(p, s - 1, m - 1), m), m); rep(i, 0, d) { ll err = a * minv(mpow(res, pe, m), m) % im; ll pos = mpow(err, mpow(p, d - 1 - i, m - 1), m); rep(j, 0, block + 1) { if (mp.count(pos)) { res = res * mpow(c, (block * mp[pos] + j) * mpow(p, i, m - 1) % (m - 1), m) % im; break; } pos = pos * gs % im; } } return res; }; for (ll d = 2; d * d <= g; d++) if (g % d == 0) { int sz = 0; while (g % d == 0) { g /= d; sz++; } a = _subroot(d, sz, a); } if (g > 1) a = _subroot(g, 1, a); return a; } ull floor_root(ull a, ull k) { if (a <= 1 or k == 1) return a; if (k >= 64) return 1; if (k == 2) return sqrtl(a); constexpr ull LIM = -1; if (a == LIM) a--; auto mul = [&](ull &x, const ull &y) { if (x <= LIM / y) x *= y; else x = LIM; }; auto pw = [&](ull x, ull t) -> ull { ull y = 1; while (t) { if (t & 1) mul(y, x); mul(x, x); t >>= 1; } return y; }; ull ret = (k == 3 ? cbrt(a) - 1 : pow(a, nextafter(1 / double(k), 0))); while (pw(ret + 1, k) <= a) ret++; return ret; } /** * @brief Primitive Function */ #line 6 "sol.cpp" ll phi(ll x) { auto ps = Pollard(x); UNIQUE(ps); ll ret = x; for (auto &p : ps) { ret /= p; ret *= p - 1; } return ret; } void solve(int _rot) { // print("Case #"+to_string(_rot)+":"); ll B, N, M; read(B, N, M); ll a; { ll n = N + 1; if (B <= 10000) chmax(n, 4LL); ll Bn = 1, BnN = 1; rep(_, 0, n) Bn *= B; rep(_, 0, n + N) BnN *= B; ll mpw = mpow(M, Bn, BnN); if (mpw % Bn != 1) { print(-1); return; } a = (mpw - 1) / Bn; } print(a); rep(nn, N + 1, N * 2) { ll Bn = 1, BnN = 1; rep(_, 0, nn) Bn *= B; rep(_, 0, nn + N) BnN *= B; ll mpw = mpow(M, Bn, BnN); if (mpw % Bn != 1) { assert(0); } ll aa = (mpw - 1) / Bn; if (a != aa) { show(a, aa); assert(0); } } } int main() { int t; read(t); rep(rot, 0, t) solve(rot + 1); return 0; }