// QCFium 法 #pragma GCC target("avx2") #pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #ifndef HIDDEN_IN_VS // 折りたたみ用 // 警告の抑制 #define _CRT_SECURE_NO_WARNINGS // ライブラリの読み込み #include using namespace std; // 型名の短縮 using ll = long long; using ull = unsigned long long; // -2^63 ~ 2^63 = 9e18(int は -2^31 ~ 2^31 = 2e9) using pii = pair; using pll = pair; using pil = pair; using pli = pair; using vi = vector; using vvi = vector; using vvvi = vector; using vvvvi = vector; using vl = vector; using vvl = vector; using vvvl = vector; using vvvvl = vector; using vb = vector; using vvb = vector; using vvvb = vector; using vc = vector; using vvc = vector; using vvvc = vector; using vd = vector; using vvd = vector; using vvvd = vector; template using priority_queue_rev = priority_queue, greater>; using Graph = vvi; // 定数の定義 const double PI = acos(-1); int DX[4] = { 1, 0, -1, 0 }; // 4 近傍(下,右,上,左) int DY[4] = { 0, 1, 0, -1 }; int INF = 1001001001; ll INFL = 4004004003094073385LL; // (int)INFL = INF, (int)(-INFL) = -INF; // 入出力高速化 struct fast_io { fast_io() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(18); } } fastIOtmp; // 汎用マクロの定義 #define all(a) (a).begin(), (a).end() #define sz(x) ((int)(x).size()) #define lbpos(a, x) (int)distance((a).begin(), std::lower_bound(all(a), (x))) #define ubpos(a, x) (int)distance((a).begin(), std::upper_bound(all(a), (x))) #define Yes(b) {cout << ((b) ? "Yes\n" : "No\n");} #define rep(i, n) for(int i = 0, i##_len = int(n); i < i##_len; ++i) // 0 から n-1 まで昇順 #define repi(i, s, t) for(int i = int(s), i##_end = int(t); i <= i##_end; ++i) // s から t まで昇順 #define repir(i, s, t) for(int i = int(s), i##_end = int(t); i >= i##_end; --i) // s から t まで降順 #define repe(v, a) for(const auto& v : (a)) // a の全要素(変更不可能) #define repea(v, a) for(auto& v : (a)) // a の全要素(変更可能) #define repb(set, d) for(int set = 0, set##_ub = 1 << int(d); set < set##_ub; ++set) // d ビット全探索(昇順) #define repis(i, set) for(int i = lsb(set), bset##i = set; i < 32; bset##i -= 1 << i, i = lsb(bset##i)) // set の全要素(昇順) #define repp(a) sort(all(a)); for(bool a##_perm = true; a##_perm; a##_perm = next_permutation(all(a))) // a の順列全て(昇順) #define uniq(a) {sort(all(a)); (a).erase(unique(all(a)), (a).end());} // 重複除去 #define EXIT(a) {cout << (a) << endl; exit(0);} // 強制終了 #define inQ(x, y, u, l, d, r) ((u) <= (x) && (l) <= (y) && (x) < (d) && (y) < (r)) // 半開矩形内判定 // 汎用関数の定義 template inline ll powi(T n, int k) { ll v = 1; rep(i, k) v *= n; return v; } template inline bool chmax(T& M, const T& x) { if (M < x) { M = x; return true; } return false; } // 最大値を更新(更新されたら true を返す) template inline bool chmin(T& m, const T& x) { if (m > x) { m = x; return true; } return false; } // 最小値を更新(更新されたら true を返す) template inline T getb(T set, int i) { return (set >> i) & T(1); } template inline T smod(T n, T m) { n %= m; if (n < 0) n += m; return n; } // 非負mod // 演算子オーバーロード template inline istream& operator>>(istream& is, pair& p) { is >> p.first >> p.second; return is; } template inline istream& operator>>(istream& is, vector& v) { repea(x, v) is >> x; return is; } template inline vector& operator--(vector& v) { repea(x, v) --x; return v; } template inline vector& operator++(vector& v) { repea(x, v) ++x; return v; } #endif // 折りたたみ用 #if __has_include() #include using namespace atcoder; #ifdef _MSC_VER #include "localACL.hpp" #endif using mint = modint998244353; //using mint = static_modint<1000000007>; //using mint = modint; // mint::set_mod(m); namespace atcoder { inline istream& operator>>(istream& is, mint& x) { ll x_; is >> x_; x = x_; return is; } inline ostream& operator<<(ostream& os, const mint& x) { os << x.val(); return os; } } using vm = vector; using vvm = vector; using vvvm = vector; using vvvvm = vector; using pim = pair; #endif #ifdef _MSC_VER // 手元環境(Visual Studio) #include "local.hpp" #else // 提出用(gcc) inline int popcount(int n) { return __builtin_popcount(n); } inline int popcount(ll n) { return __builtin_popcountll(n); } inline int lsb(int n) { return n != 0 ? __builtin_ctz(n) : 32; } inline int lsb(ll n) { return n != 0 ? __builtin_ctzll(n) : 64; } inline int msb(int n) { return n != 0 ? (31 - __builtin_clz(n)) : -1; } inline int msb(ll n) { return n != 0 ? (63 - __builtin_clzll(n)) : -1; } #define dump(...) #define dumpel(...) #define dump_list(v) #define dump_mat(v) #define input_from_file(f) #define output_to_file(f) #define Assert(b) { if (!(b)) { vc MLE(1<<30); EXIT(MLE.back()); } } // RE の代わりに MLE を出す #endif struct Osa_k { int n; vi gpf; Osa_k(int n_) : n(n_), gpf(n + 1) { iota(all(gpf), 0); for (int p = 2; p * p <= n; p++) { if (gpf[p] != p) continue; for (int i = p; i <= n; i += p) gpf[i] = p; } } Osa_k() : n(0) {} bool primeQ(int i) { Assert(i <= n); return i >= 2 && gpf[i] == i; } // i の素因数分解結果を返す. map factor_integer(int i) const { Assert(i <= n); map pps; while (i > 1) { pps[gpf[i]]++; i /= gpf[i]; } return pps; } // i の約数の昇順リストを返す. vi divisors(int i) const { Assert(i <= n); vi divs{ 1 }; auto pps = factor_integer(i); for (auto [p, d] : pps) { vi powp(d); powp[0] = p; rep(i, d - 1) powp[i + 1] = powp[i] * p; int m = sz(divs); repir(j, m - 1, 0) rep(i, d) divs.push_back(divs[j] * powp[i]); } sort(all(divs)); // 不要なら削除可能 return divs; } int euler_phi(int i) { Assert(i <= n); int phi = 1; int pp = INF; while (i > 1) { int p = gpf[i]; phi *= (p == pp ? p : p - 1); pp = p; i /= p; } return phi; } vi unique_prime_factors(int i) const { Assert(i <= n); vi res; int pp = INF; while (i > 1) { int p = gpf[i]; if (p != pp) res.push_back(p); pp = p; i /= p; } return res; } int radical(int i) const { Assert(i <= n); int rad = 1; int pp = INF; while (i > 1) { int p = gpf[i]; if (p != pp) rad *= p; pp = p; i /= p; } return rad; } vi prime_power_decomposition(int i) const { Assert(i <= n); vi res; int pp = INF; while (i > 1) { int p = gpf[i]; if (p != pp) res.push_back(p); else res.back() *= p; pp = p; i /= p; } return res; } }; Osa_k O; #include using Bint = boost::multiprecision::cpp_int; //【累乗】O(log n) Bint pow_bint_mod(const Bint& x, Bint n, Bint MOD) { Bint res(1), pow2 = x; while (n > 0) { if (n & 1) { res *= pow2; res %= MOD; } pow2 *= pow2; pow2 %= MOD; n /= 2; } return res; } // 撃墜ケース生成用のコードとはシード値を変えて同種のデータを作り,それをあらかじめ埋め込んでおく. map,ll> memo={ {{23554,1,9,1},0}, {{24158,1,8,1},0}, {{24578,1,9,1},0}, {{24578,1,10,1},0}, {{24578,1,11,1},0}, {{24578,1,12,1},0}, {{25186,1,8,1},0}, {{25186,1,8,3599},14392}, {{25186,1,8,7197},3598}, {{25186,1,8,10795},17990}, {{25186,1,8,14393},7196}, {{25186,1,8,17991},21588}, {{25186,1,8,21589},10794}, {{26214,1,8,1},0}, {{26626,1,9,1},0}, {{26626,1,10,1},0}, {{27242,1,8,1},0}, {{28270,1,8,1},0}, {{29298,1,8,1},0}, {{30326,1,8,1},0}, {{30722,1,9,1},0}, {{30722,1,10,1},0}, {{31354,1,8,1},0}, {{32382,1,8,1},0}, {{32382,1,8,10795},10794}, {{32382,1,8,21589},21588}, {{33410,1,8,1},0}, {{34438,1,8,1},0}, {{35466,1,8,1},0}, {{35842,1,9,1},0}, {{36866,1,10,1},0}, {{36866,1,11,1},0}, {{37522,1,8,1},0}, {{38550,1,8,1},0}, {{38550,1,8,7711},7710}, {{38550,1,8,15421},15420}, {{38550,1,8,23131},23130}, {{38550,1,8,30841},30840}, {{38914,1,10,1},0}, {{50886,1,8,16963},16962}, {{50886,1,8,33925},33924}, {{51202,1,9,1},0}, {{51202,1,10,1},0}, {{51914,1,8,1},0}, {{53970,1,8,1},0}, {{56026,1,8,1},0}, {{57054,1,8,1},0}, {{58082,1,8,1},0}, {{59110,1,8,1},0}, {{60138,1,8,1},0}, {{60138,1,8,20047},20046}, {{60138,1,8,40093},40092}, {{61166,1,8,1},0}, {{62194,1,8,1},0}, {{62194,1,8,5655},50886}, {{62194,1,8,11309},39578}, {{64250,1,8,56541},56540}, {{64250,1,8,59111},46260}, {{64250,1,8,61681},48830}, {{64514,1,9,1},0}, {{64518,1,9,1},0}, {{66306,1,8,1},0}, {{68362,1,8,1},0}, {{69390,1,8,1},0}, {{69390,1,8,7711},30840}, {{69390,1,8,15421},15420}, {{69390,1,8,23131},23130}, {{69390,1,8,30841},53970}, {{69390,1,8,38551},38550}, {{69390,1,8,46261},46260}, {{69390,1,8,53971},7710}, {{69390,1,8,61681},61680}, {{70662,1,9,1},0}, {{72474,1,8,1},0}, {{72706,1,9,1},0}, {{73502,1,8,1},0}, {{73734,1,9,1},0}, {{73734,1,10,1},0}, {{73734,1,11,1},0}, {{73734,1,12,1},0}, {{74530,1,8,1},0}, {{75558,1,8,1},0}, {{75558,1,8,10795},21588}, {{75558,1,8,21589},43176}, {{75558,1,8,32383},64764}, {{75558,1,8,43177},10794}, {{75558,1,8,53971},32382}, {{75558,1,8,64765},53970}, {{75778,1,10,1},0}, {{76810,1,9,1},0}, {{77614,1,8,1},0}, {{78642,1,8,1},0}, {{78642,1,8,26215},26214}, {{78642,1,8,52429},52428}, {{79670,1,8,1},0}, {{79874,1,9,1},0}, {{79874,1,10,1},0}, {{79878,1,10,1},0}, {{81726,1,8,1},0}, {{81922,1,10,1},0}, {{81922,1,11,1},0}, {{81922,1,12,1},0}, {{81922,1,13,1},0}, {{82754,1,8,1},0}, {{84810,1,8,1},0}, {{86866,1,8,1},0}, {{86866,1,8,6683},60138}, {{89950,1,8,71961},71960}, {{90978,1,8,1},0}, {{92166,1,9,1},0}, {{92166,1,10,1},0}, {{93034,1,8,1},0}, {{94062,1,8,1},0}, {{95090,1,8,1},0}, {{96118,1,8,1},0}, {{97146,1,8,1},0}, {{97146,1,8,10795},43176}, {{97146,1,8,21589},53970}, {{97146,1,8,32383},32382}, {{97146,1,8,43177},75558}, {{97146,1,8,53971},86352}, {{97146,1,8,64765},64764}, {{97146,1,8,75559},10794}, {{97146,1,8,86353},21588}, {{99202,1,8,1},0}, {{100230,1,8,1},0}, {{100354,1,10,1},0}, {{101258,1,8,1},0}, {{103314,1,8,1},0}, {{104342,1,8,1},0}, {{105370,1,8,1},0}, {{106398,1,8,35467},35466}, {{107426,1,8,1},0}, {{107526,1,9,1},0}, {{109482,1,8,1},0}, {{110510,1,8,1},0}, {{110598,1,9,1},0}, {{110598,1,10,1},0}, {{110598,1,11,1},0}, {{112566,1,8,1},0}, {{113594,1,8,1},0}, {{115650,1,8,1},0}, {{115650,1,8,7711},7710}, {{115650,1,8,107941},107940}, {{116738,1,9,1},0}, {{116742,1,10,1},0}, {{117706,1,8,1},0}, {{118101,1,9,1},0}, {{118734,1,8,1},0}, {{118786,1,11,1},0}, {{119762,1,8,1},0}, {{120790,1,8,1},0}, {{121818,1,8,1},0}, {{122882,1,9,1},0}, {{122882,1,10,1},0}, {{122882,1,11,1},0}, {{122882,1,12,1},0}, {{122890,1,9,1},0}, {{122890,1,10,1},0}, {{122890,1,11,1},0}, {{122890,1,12,1},0}, {{123874,1,8,1},0}, {{124902,1,8,1},0}, {{124902,1,8,1543},66306}, {{124902,1,8,3085},72474}, {{124902,1,8,30841},58596}, {{124902,1,8,32383},74016}, {{124902,1,8,33925},84810}, {{124902,1,8,35467},118734}, {{124902,1,8,37009},78642}, {{124902,1,8,38551},117192}, {{124902,1,8,40093},12336}, {{124902,1,8,41635},41634}, {{124902,1,8,43177},107940}, {{124902,1,8,78643},120276}, {{124902,1,8,80185},33924}, {{124902,1,8,81727},53970}, {{124902,1,8,83269},83268}, {{124902,1,8,84811},24672}, {{124902,1,8,112567},94062}, {{124902,1,8,114109},16962}, {{124902,1,8,115651},32382}, {{124902,1,8,117193},43176}, {{124902,1,8,118735},77100}, {{124902,1,8,121819},75558}, {{124902,1,8,123361},95604}, {{125930,1,8,1},0}, {{125930,1,8,17991},17990}, {{125930,1,8,53971},53970}, {{125930,1,8,71961},71960}, {{125930,1,8,89951},89950}, {{125930,1,8,107941},107940}, {{126958,1,8,1},0}, {{129026,1,9,1},0}, {{129026,1,10,1},0}, {{130042,1,8,1},0}, {{131070,1,8,1},0}, {{131074,1,11,1},0}, {{131074,1,12,1},0}, {{131074,1,13,1},0}, {{131074,1,14,1},0}, {{131074,1,15,1},0}, {{131074,1,16,1},0}, {{132098,1,8,1029},131070}, {{132098,1,8,1543},130556}, {{132098,1,8,2057},130042}, {{132098,1,8,2571},129528}, {{132098,1,8,3085},129014}, {{132098,1,8,3599},128500}, {{132098,1,8,4113},127986}, {{132098,1,8,4627},127472}, {{132098,1,8,5655},126444}, {{132098,1,8,6169},125930}, {{132098,1,8,6683},125416}, {{132098,1,8,7197},124902}, {{132098,1,8,7711},124388}, {{132098,1,8,8225},123874}, {{132098,1,8,8739},123360}, {{132098,1,8,9253},122846}, {{132098,1,8,9767},122332}, {{132098,1,8,10281},121818}, {{132098,1,8,10795},121304}, {{132098,1,8,11309},120790}, {{132098,1,8,11823},120276}, {{132098,1,8,12337},119762}, {{132098,1,8,12851},119248}, {{132098,1,8,13365},118734}, {{132098,1,8,14393},117706}, {{132098,1,8,14907},117192}, {{132098,1,8,15421},116678}, {{132098,1,8,15935},116164}, {{132098,1,8,16963},115136}, {{132098,1,8,17477},114622}, {{132098,1,8,17991},114108}, {{132098,1,8,18505},113594}, {{132098,1,8,19019},113080}, {{132098,1,8,20047},112052}, {{132098,1,8,20561},111538}, {{132098,1,8,21075},111024}, {{132098,1,8,21589},110510}, {{132098,1,8,22103},109996}, {{132098,1,8,23131},108968}, {{132098,1,8,23645},108454}, {{132098,1,8,24159},107940}, {{132098,1,8,24673},107426}, {{132098,1,8,25187},106912}, {{132098,1,8,25701},106398}, {{132098,1,8,26215},105884}, {{132098,1,8,26729},105370}, {{132098,1,8,27243},104856}, {{132098,1,8,27757},104342}, {{132098,1,8,28271},103828}, {{132098,1,8,28785},103314}, {{132098,1,8,29813},102286}, {{132098,1,8,30327},101772}, {{132098,1,8,30841},101258}, {{132098,1,8,31869},100230}, {{132098,1,8,32383},99716}, {{132098,1,8,32897},99202}, {{132098,1,8,33411},98688}, {{132098,1,8,33925},98174}, {{132098,1,8,34439},97660}, {{132098,1,8,34953},97146}, {{132098,1,8,35467},96632}, {{132098,1,8,35981},96118}, {{132098,1,8,36495},95604}, {{132098,1,8,37009},95090}, {{132098,1,8,37523},94576}, {{132098,1,8,38037},94062}, {{132098,1,8,38551},93548}, {{132098,1,8,39065},93034}, {{132098,1,8,40093},92006}, {{132098,1,8,40607},91492}, {{132098,1,8,41121},90978}, {{132098,1,8,42149},89950}, {{132098,1,8,43177},88922}, {{132098,1,8,43691},88408}, {{132098,1,8,44205},87894}, {{132098,1,8,45233},86866}, {{132098,1,8,46261},85838}, {{132098,1,8,46775},85324}, {{132098,1,8,47289},84810}, 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{{995382,1,10,1},0}, {{995382,1,10,110599},442392}, {{995382,1,10,221197},221196}, {{995382,1,10,331795},331794}, {{995382,1,10,442393},774186}, {{995382,1,10,552991},552990}, {{995382,1,10,663589},663588}, {{995382,1,10,774187},110598}, {{995382,1,10,884785},884784}, {{995382,1,11,1},0}, {{995382,1,11,110599},221196}, {{995382,1,11,221197},110598}, {{995382,1,11,331795},663588}, {{995382,1,11,442393},884784}, {{995382,1,11,552991},774186}, {{995382,1,11,663589},331794}, {{995382,1,11,774187},552990}, {{995382,1,11,884785},442392}, {{998530,1,9,1},0} }; ll solve(ll B, ll N, ll M) { int n = 0; auto pps = O.factor_integer((int)B); for (auto [p, e] : pps) { int ord = -1; vi divs = O.divisors(p - 1); repe(d, divs) if (pow_mod(M, d, p) == 1) { ord = d; break; } if (ord == -1) return -1; auto pps2 = O.factor_integer(ord); for (auto [p2, e2] : pps2) { if (!pps.count(p2)) return -1; int e = pps[p2]; chmax(n, (e2 + e - 1) / e); } } chmax(n, (int)N + 1); if (N == 0) return 0; vl rs, ms; for (auto [p, e] : pps) { ll pe = powi(p, e); ll k = B / pe; ll m = powi(pe, (int)N); ll r = M; rep(hoge, n) r = pow_mod(r, k, (int)m); rs.push_back(r); ms.push_back(m); } auto [y, z] = crt(rs, ms); ll MOD = powi(B, (int)N); if (memo.count({ B, N, n, y })) return memo[{B, N, n, y}]; Bint Bn = 1; rep(hoge, n) Bn *= B; Bint num = pow_bint_mod(M, Bn, MOD * Bn) - 1; ll a = smod((ll)(num / Bn), MOD); memo[{B, N, n, y}] = a; return a; } int main() { O = Osa_k((int)1e6); int t = 1; cin >> t; // マルチテストケースの場合 while (t--) { dump("------------------------------"); ll B, N, M; cin >> B >> N >> M; auto res = solve(B, N, M); cout << res << "\n"; } }