using System; using System.Collections; using System.Collections.Generic; using System.Diagnostics; using System.IO; using System.Linq; using System.Numerics; using System.Text; using System.Text.RegularExpressions; using System.Threading.Tasks; using static System.Math; using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute; using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions; public static class P { public static void Main() { Action abort = () => { Console.WriteLine(-1); Environment.Exit(0); }; var line = Console.ReadLine(); Assert(Regex.IsMatch(line, @"^\d+ \d+ \d+$")); var pka = line.Split().Select(int.Parse).ToArray(); var p = pka[0]; var k = pka[1]; var a = pka[2]; Assert(2 <= k && k <= 1000); Assert(2 <= p && p <= 1000000007); Assert(1 <= a && a < p); ModInt.Mod = p; //個人的な興味で離散対数のアルゴリズムをポラードローにしています //有限体Z/pZの原始根gを一つ求め、 var g = GetPrimitiveRoot(p); //そのgについてのaの指数を求める。 var aIndex = Log(a, g); Assert(aIndex != 0); //これによって、Z/(p-1)Z上でのa/kを求める問題に帰着できる //∵x^k≡a(mod p) ⇔ (g^xInd)^k≡(g^aInd)(mod p) ⇔ (xInd)*k≡aInd(mod p-1) //x=a/kとすると、kx=aより、aはZ/(p-1)Z上でik(i∈ℕ)と等しい必要がある。 //これは、Z/(p-1)Z上において指数がgcd(k, p - 1)で割り切れる要素のみで構成される群の要素と等しい。 var step = GCD(k, p - 1); //よって、stepを生成元として(p-1)を法とする加法によって生成される巡回群の位数orderは、 var order = (p - 1) / step; //まず、aがその巡回群に乗っていない場合、除算は不可能。 if (aIndex % step != 0) abort(); //Z/orderZ上でのa,kの指数はそれぞれ、 var aIndexOnZOrderZ = aIndex / step; var kIndexOnZOrderZ = k / step; //ここで、Z/orderZ上でのa/kの指数は var aPerKIndex = (aIndexOnZOrderZ * GetInverse(kIndexOnZOrderZ, order)) % order; //Z/orderZはZ/(p-1)Zの部分群なので、Z/(p-1)Zにおいても指数は等しい。よって、 Console.WriteLine(Power(g, aPerKIndex)); } static int GetPrimitiveRoot(long m) { var subgroupOrders = new List(); var order = m - 1; if ((order & 1) == 0) { while ((order & 1) == 0) order >>= 1; subgroupOrders.Add((m - 1) / 2); } for (long i = 3; i * i <= order; i += 2) if (order % i == 0) { while (order % i == 0) order /= i; subgroupOrders.Add((m - 1) / i); } if (order != 1) subgroupOrders.Add((m - 1) / order); for (int g = 2; g < m; g++) { if (subgroupOrders.Any(x => Power(g, x) == 1)) continue; return g; } throw new Exception(); } static int Log(ModInt a, ModInt g) { if (a == 1) return 0; long mulOrder = ModInt.Mod - 1; int tryCount = 1; var curG = g; while (true) { //Console.WriteLine($"Initializing {tryCount}th attempt..."); ModInt x = 1; long xAInd = 0; long xGInd = 0; ModInt waiting = 1; long waitingAInd = 0; long waitingGInd = 0; int step = 9; while (true) { for (int i = 0; i < step; i += 3) { x *= curG; xGInd += tryCount; //Console.WriteLine($"x = {x} = a^{xAInd} * g^{xGInd}"); if (x == waiting) goto found; x *= x; xAInd = (xAInd << 1) % mulOrder; xGInd = (xGInd << 1) % mulOrder; //Console.WriteLine($"x = {x} = a^{xAInd} * g^{xGInd}"); if (x == waiting) goto found; x *= a; xAInd++; //Console.WriteLine($"x = {x} = a^{xAInd} * g^{xGInd}"); if (x == waiting) goto found; } step <<= 1; waiting = x; waitingAInd = xAInd; waitingGInd = xGInd; } found:; //Console.WriteLine("Loop found."); var aInd = (xAInd - waitingAInd + mulOrder) % mulOrder; var gInd = (waitingGInd - xGInd + mulOrder) % mulOrder; if (aInd == 0 || gInd == 0) { /*Console.WriteLine("failed.");*/ curG *= g; tryCount++; continue; } var gcd = GCD(mulOrder, aInd); if (gInd % gcd != 0) return -1; var subGroupOrder = mulOrder / gcd; return (int)(((gInd / gcd) * GetInverse(aInd / gcd, subGroupOrder)) % subGroupOrder); } } static ModInt Power(ModInt n, long m) { ModInt pow = n; ModInt res = 1; while (m > 0) { if ((m & 1) == 1) res *= pow; pow *= pow; m >>= 1; } return res; } static void Assert(bool cond) { if (!cond) throw new Exception(); } static long GCD(long a, long b) { while (true) { if (b == 0) return a; a %= b; if (a == 0) return b; b %= a; } } static long GetInverse(long a, long MOD) { long div, p = MOD, x1 = 1, y1 = 0, x2 = 0, y2 = 1; while (true) { if (p == 1) return x2 + MOD; div = a / p; x1 -= x2 * div; y1 -= y2 * div; a %= p; if (a == 1) return x1 + MOD; div = p / a; x2 -= x1 * div; y2 -= y1 * div; p %= a; } } } struct ModInt { public static int Mod { get { return MOD; } set { MOD = value; POSITIVIZER = (long)MOD << 31; } } static int MOD = 1000000007; static long POSITIVIZER = ((long)MOD) << 31; long Data; public ModInt(long data) { if ((Data = data % MOD) < 0) Data += MOD; } public static implicit operator long(ModInt modInt) => modInt.Data; public static implicit operator ModInt(long val) => new ModInt(val); public static ModInt operator +(ModInt a, int b) => new ModInt() { Data = (a.Data + b + POSITIVIZER) % MOD }; public static ModInt operator +(ModInt a, long b) => new ModInt(a.Data + b); public static ModInt operator +(ModInt a, ModInt b) { long res = a.Data + b.Data; return new ModInt() { Data = res >= MOD ? res - MOD : res }; } public static ModInt operator -(ModInt a, int b) => new ModInt() { Data = (a.Data - b + POSITIVIZER) % MOD }; public static ModInt operator -(ModInt a, long b) => new ModInt(a.Data - b); public static ModInt operator -(ModInt a, ModInt b) { long res = a.Data - b.Data; return new ModInt() { Data = res < 0 ? res + MOD : res }; } public static ModInt operator *(ModInt a, int b) => new ModInt(a.Data * b); public static ModInt operator *(ModInt a, long b) => a * new ModInt(b); public static ModInt operator *(ModInt a, ModInt b) => new ModInt() { Data = a.Data * b.Data % MOD }; public static ModInt operator /(ModInt a, ModInt b) => new ModInt() { Data = a.Data * GetInverse(b) % MOD }; public override string ToString() => Data.ToString(); static long GetInverse(long a) { long div, p = MOD, x1 = 1, y1 = 0, x2 = 0, y2 = 1; while (true) { if (p == 1) return x2 + MOD; div = a / p; x1 -= x2 * div; y1 -= y2 * div; a %= p; if (a == 1) return x1 + MOD; div = p / a; x2 -= x1 * div; y2 -= y1 * div; p %= a; } } }