import std.conv, std.functional, std.range, std.stdio, std.string; import std.algorithm, std.array, std.bigint, std.bitmanip, std.complex, std.container, std.math, std.mathspecial, std.numeric, std.regex, std.typecons; import core.bitop; class EOFException : Throwable { this() { super("EOF"); } } string[] tokens; string readToken() { for (; tokens.empty; ) { if (stdin.eof) { throw new EOFException; } tokens = readln.split; } auto token = tokens.front; tokens.popFront; return token; } int readInt() { return readToken.to!int; } long readLong() { return readToken.to!long; } real readReal() { return readToken.to!real; } bool chmin(T)(ref T t, in T f) { if (t > f) { t = f; return true; } else { return false; } } bool chmax(T)(ref T t, in T f) { if (t < f) { t = f; return true; } else { return false; } } int binarySearch(alias pred, T)(in T[] as) { int lo = -1, hi = cast(int)(as.length); for (; lo + 1 < hi; ) { const mid = (lo + hi) >> 1; (unaryFun!pred(as[mid]) ? hi : lo) = mid; } return hi; } int lowerBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a >= val)); } int upperBound(T)(in T[] as, T val) { return as.binarySearch!(a => (a > val)); } // floor(a^(1/k)) ulong floorKthRoot(ulong a, ulong k) { import core.bitop : bsr; import std.algorithm : min; if (a == 0) { return 0; } else if (k <= 1) { return a; } else if (k == 2) { ulong b = a, x = 0, y = 0; for (int e = bsr(a) & ~1; e >= 0; e -= 2) { x <<= 1; y <<= 1; if (b >= (y | 1) << e) { b -= (y | 1) << e; x |= 1; y += 2; } } return x; } else if (k <= 40) { // min x s.t. x^k >= 2^64 enum ulong[] HIS = [0, 0, 4294967296UL, 2642246, 65536, 7132, 1626, 566, 256, 139, 85, 57, 41, 31, 24, 20, 16, 14, 12, 11, 10, 9, 8, 7, 7, 6, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4]; ulong lo = 1UL << (bsr(a) / k); ulong hi = min(1UL << (bsr(a) / k + 1), HIS[cast(size_t)(k)]); for (; lo + 1 < hi; ) { const ulong mid = (lo + hi) / 2; ulong b = mid * mid; foreach (i; 2 .. k) b *= mid; ((b <= a) ? lo : hi) = mid; } return lo; } else if (k <= 63) { return ((1UL << k) <= a) ? 2 : 1; } else { return 1; } } // xorshift uint xrand() { static uint x = 314159265, y = 358979323, z = 846264338, w = 327950288; uint t = x ^ x << 11; x = y; y = z; z = w; return w = w ^ w >> 19 ^ t ^ t >> 8; } // a^-1 (mod m) long modInv(long a, long m) in { assert(m > 0, "modInv: m > 0 must hold"); } do { long b = m, x = 1, y = 0, t; for (; ; ) { t = a / b; a -= t * b; if (a == 0) { assert(b == 1 || b == -1, "modInv: gcd(a, m) != 1"); if (b == -1) y = -y; return (y < 0) ? (y + m) : y; } x -= t * y; t = b / a; b -= t * a; if (b == 0) { assert(a == 1 || a == -1, "modInv: gcd(a, m) != 1"); if (a == -1) x = -x; return (x < 0) ? (x + m) : x; } y -= t * x; } } // (a / 2) mod m long div2(long a, long m) in { assert(1 <= m && m < 1L << 62, "div2: 1 <= m < 2^62 must hold"); assert(m % 2 != 0, "div2: m must not be divisibly by 3"); assert(0 <= a && a < m, "div2: 0 <= a < m must hold"); } do { return (a + (a % 2) * m) / 2; } // (a / 3) mod m long div3(long a, long m) in { assert(1 <= m && m < 1L << 61, "div3: 1 <= m < 2^61 must hold"); assert(m % 3 != 0, "div3: m must not be divisibly by 3"); assert(0 <= a && a < m, "div3: 0 <= a < m must hold"); } do { return (a + (((3 - a % 3) * (m % 3)) % 3) * m) / 3; } // Jacobi symbol (a/m) int jacobi(long a, long m) in { assert(m > 0, "jacobi: m > 0 must hold"); assert(m & 1, "jacobi: m must be odd"); } do { import core.bitop : bsf; import std.algorithm.mutation : swap; int s = 1; if (a < 0) a = a % m + m; for (; m > 1; ) { a %= m; if (a == 0) return 0; const r = bsf(a); if ((r & 1) && ((m + 2) & 4)) s = -s; a >>= r; if (a & m & 2) s = -s; swap(a, m); } return s; } // sqrt(a) (mod p) // p: be prime // (b + sqrt(b^2 - a))^((p+1)/2) in F_p(sqrt(b^2 - a)) long[] modSqrt(long a, long p) in { assert(p < 1L << 31, "modSqrt: p < 2^31 must hold"); assert(-p * p <= a && a <= p * p, "modSqrt: -p^2 <= a <= p^2 must hold"); } do { if (p == 2) return [a & 1]; const j = jacobi(a, p); if (j == 0) return [0]; if (j == -1) return []; long b, d; for (; ; ) { b = xrand() % p; d = (b * b - a) % p; if (d < 0) d += p; if (jacobi(d, p) == -1) break; } long[2] mul(in long[2] x, in long[2] y) { return [(x[0] * y[0] + d * ((x[1] * y[1]) % p)) % p, (x[0] * y[1] + x[1] * y[0]) % p]; } long[2] f = [b, 1], g = [1, 0]; for (long e = (p + 1) >> 1; e; e >>= 1) { if (e & 1) g = mul(g, f); f = mul(f, f); } assert(g[1] == 0); return (g[0] < p - g[0]) ? [g[0], p - g[0]] : [p - g[0], g[0]]; } // Roots of f0 + f1 T + T^2 in F_p[T] with multiplicity // p: prime long[] modRoots2(long f0, long f1, long p) in { assert(2 <= p && p < 1L << 31, "modRoots2: 2 <= p < 2^31 must hold"); assert(0 <= f0 && f0 < p, "modRoots2: 0 <= f0 < p must hold"); assert(0 <= f1 && f1 < p, "modRoots2: 0 <= f1 < p must hold"); } do { import std.algorithm : sort; if (p == 2) { if (f0 == 0 && f1 == 0) return [0, 0]; if (f0 == 0 && f1 == 1) return [0, 1]; if (f0 == 1 && f1 == 0) return [1, 1]; return []; } else { const f12 = f1.div2(p); auto ts = modSqrt(f12 * f12 - f0, p); foreach (ref t; ts) { if ((t -= f12) < 0) t += p; } sort(ts); switch (ts.length) { case 0: return []; case 1: return [ts[0], ts[0]]; case 2: return ts; default: assert(false); } } } Int mod(Int)(Int a, Int m) { if ((a %= m) < 0) a += m; return a; } Int gcd(Int)(Int a, Int b) { return (b != 0) ? gcd(b, a % b) : a; } Int lcm(Int)(Int a, Int b) { return a / gcd(a, b) * b; } Int gojo(Int)(Int a, Int b, out Int x, out Int y) { if (b != 0) { Int g = gojo(b, a % b, y, x); y -= (a / b) * x; return g; } x = 1; y = 0; return a; } Int modInv(Int)(Int a, Int m) { Int x, y; gojo(a, m, x, y); return mod(x, m); } long modLogP(long a, long b, long m) { a = mod(a, m); b = mod(b, m); if (m == 1) return 0; long k, al = 1; Tuple!(long,long)[] as; for (k = 0; k * k < m; ++k) { as ~= tuple(al, k); al = (al * a) % m; } as.sort; al = modInv(al, m); for (long i = 0; i < k; ++i) { int pos = as.lowerBound(tuple(b, 0L)); if (pos < as.length && as[pos][0] == b) return i * k + as[pos][1]; b = (b * al) % m; } return -1; } long P; long[][] A, B; long[][] div(long[][] a, long t) { const invT = modInv(t, P); auto b = new long[][](2, 2); foreach (i; 0 .. 2) foreach (j; 0 .. 2) { b[i][j] = mod(invT * a[i][j], P); } return b; } long[][] inv(long[][] a) { const det = mod(a[0][0] * a[1][1] - a[0][1] * a[1][0], P); return div([[a[1][1], -a[0][1]], [-a[1][0], a[0][0]]], det); } long[][] mul(long[][] a, long[][] b) { auto c = new long[][](2, 2); foreach (i; 0 .. 2) foreach (k; 0 .. 2) foreach (j; 0 .. 2) { c[i][j] = mod(c[i][j] + a[i][k] * b[k][j], P); } return c; } long[][] power(long[][] a, long e) { long[][] b = a, c = [[1, 0], [0, 1]]; for (; e; e >>= 1) { if (e & 1) c = mul(c, b); b = mul(b, b); } return c; } long solveBSGS(long[][] a, long[][] b) { const m = cast(int)(floorKthRoot(2 * P, 2) + 2); auto als = new Tuple!(long[][], int)[m]; long[][] al = [[1, 0], [0, 1]]; foreach (l; 0 .. m) { als[l] = tuple(al, l); al = mul(al, a); } als.sort; al = inv(al); long[][] amk = b; foreach (k; 0 .. m) { const pos = als.lowerBound(tuple(amk, 0)); if (pos < m && als[pos][0] == amk) { return k * cast(long)(m) + als[pos][1]; } amk = mul(amk, al); } return -1; } void main() { try { for (; ; ) { P = readLong(); A = new long[][](2, 2); foreach (i; 0 .. 2) foreach (j; 0 .. 2) { A[i][j] = readLong(); } B = new long[][](2, 2); foreach (i; 0 .. 2) foreach (j; 0 .. 2) { B[i][j] = readLong(); } long ans; const detA = mod(A[0][0] * A[1][1] - A[0][1] * A[1][0], P); const detB = mod(B[0][0] * B[1][1] - B[0][1] * B[1][0], P); if (detA == 0) { if (detB == 0) { // ? import core.stdc.stdlib; exit(1); } else { ans = -1; } } else { const period = modLogP(detA, modInv(detA, P), P) + 1; long e0 = modLogP(detA, detB, P); debug { writeln("detA = ", detA, ", detB = ", detB); writeln("period = ", period); writeln("e0 = ", e0); } if (e0 == -1) { ans = -1; } else { if (e0 == 0) { e0 += period; } auto a = power(A, period); auto b = mul(B, inv(power(A, e0))); debug { writeln("a = ", a); writeln("b = ", b); } const res = solveBSGS(a, b); if (res == -1) { ans = -1; } else { ans = e0 + period * res; } } } writeln(ans); } } catch (EOFException e) { } }