#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #ifdef RUTHEN_LOCAL #include #else #define show(x) true #endif // type definition using i64 = long long; using u32 = unsigned int; using u64 = unsigned long long; using f32 = float; using f64 = double; using f128 = long double; template using pque = std::priority_queue; template using pqueg = std::priority_queue, std::greater>; // overload #define overload4(_1, _2, _3, _4, name, ...) name #define overload3(_1, _2, _3, name, ...) name #define overload2(_1, _2, name, ...) name // for loop #define REP1(a) for (long long _ = 0; _ < (a); _++) #define REP2(i, a) for (long long i = 0; i < (a); i++) #define REP3(i, a, b) for (long long i = (a); i < (b); i++) #define REP4(i, a, b, c) for (long long i = (a); i < (b); i += (c)) #define REP(...) overload4(__VA_ARGS__, REP4, REP3, REP2, REP1)(__VA_ARGS__) #define RREP1(a) for (long long _ = (a)-1; _ >= 0; _--) #define RREP2(i, a) for (long long i = (a)-1; i >= 0; i--) #define RREP3(i, a, b) for (long long i = (b)-1; i >= (a); i--) #define RREP(...) overload3(__VA_ARGS__, RREP3, RREP2, RREP1)(__VA_ARGS__) #define FORE1(x, a) for (auto&& x : a) #define FORE2(x, y, a) for (auto&& [x, y] : a) #define FORE3(x, y, z, a) for (auto&& [x, y, z] : a) #define FORE(...) overload4(__VA_ARGS__, FORE3, FORE2, FORE1)(__VA_ARGS__) #define FORSUB(t, s) for (long long t = (s); t; t = (t - 1) & (s)) // function #define ALL(a) (a).begin(), (a).end() #define RALL(a) (a).rbegin(), (a).rend() #define SORT(a) std::sort((a).begin(), (a).end()) #define RSORT(a) std::sort((a).rbegin(), (a).rend()) #define REV(a) std::reverse((a).begin(), (a).end()) #define UNIQUE(a) \ std::sort((a).begin(), (a).end()); \ (a).erase(std::unique((a).begin(), (a).end()), (a).end()) #define LEN(a) (int)((a).size()) #define MIN(a) *std::min_element((a).begin(), (a).end()) #define MAX(a) *std::max_element((a).begin(), (a).end()) #define SUM1(a) std::accumulate((a).begin(), (a).end(), 0LL) #define SUM2(a, x) std::accumulate((a).begin(), (a).end(), (x)) #define SUM(...) overload2(__VA_ARGS__, SUM2, SUM1)(__VA_ARGS__) #define LB(a, x) std::distance((a).begin(), std::lower_bound((a).begin(), (a).end(), (x))) #define UB(a, x) std::distance((a).begin(), std::upper_bound((a).begin(), (a).end(), (x))) template inline bool chmin(T& a, const U& b) { return (a > T(b) ? a = b, 1 : 0); } template inline bool chmax(T& a, const U& b) { return (a < T(b) ? a = b, 1 : 0); } template inline T floor(const T x, const S y) { assert(y); return (y < 0 ? floor(-x, -y) : (x > 0 ? x / y : x / y - (x % y == 0 ? 0 : 1))); } template inline T ceil(const T x, const S y) { assert(y); return (y < 0 ? ceil(-x, -y) : (x > 0 ? (x + y - 1) / y : x / y)); } template std::pair inline divmod(const T x, const S y) { T q = floor(x, y); return {q, x - q * y}; } // 10 ^ n constexpr long long TEN(int n) { return (n == 0) ? 1 : 10LL * TEN(n - 1); } // 1 + 2 + ... + n #define TRI1(n) ((n) * ((n) + 1LL) / 2) // l + (l + 1) + ... + r #define TRI2(l, r) (((l) + (r)) * ((r) - (l) + 1LL) / 2) #define TRI(...) overload2(__VA_ARGS__, TRI2, TRI1)(__VA_ARGS__) // bit operation // bit[i] (= 0 or 1) #define IBIT(bit, i) (((bit) >> (i)) & 1) // (0, 1, 2, 3, 4) -> (0, 1, 3, 7, 15) #define MASK(n) ((1LL << (n)) - 1) #define POW2(n) (1LL << (n)) // (0, 1, 2, 3, 4) -> (0, 1, 1, 2, 1) int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(i64 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(i64 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 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(i64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } // binary search template T bin_search(T ok, T ng, F& f) { while ((ok > ng ? ok - ng : ng - ok) > 1) { T md = (ng + ok) >> 1; (f(md) ? ok : ng) = md; } return ok; } template T bin_search_real(T ok, T ng, F& f, const int iter = 100) { for (int _ = 0; _ < iter; _++) { T md = (ng + ok) / 2; (f(md) ? ok : ng) = md; } return ok; } // rotate matrix counterclockwise by pi / 2 template void rot(std::vector>& a) { if ((int)(a.size()) == 0) return; if ((int)(a[0].size()) == 0) return; int n = (int)(a.size()), m = (int)(a[0].size()); std::vector res(m, std::vector(n)); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { res[m - 1 - j][i] = a[i][j]; } } a.swap(res); } // const value constexpr int dx[8] = {1, 0, -1, 0, 1, -1, -1, 1}; constexpr int dy[8] = {0, 1, 0, -1, 1, 1, -1, -1}; // infinity template constexpr T INF = 0; template <> constexpr int INF = 1'000'000'000; // 1e9 template <> constexpr i64 INF = i64(INF) * INF * 2; // 2e18 template <> constexpr u32 INF = INF; // 1e9 template <> constexpr u64 INF = INF; // 2e18 template <> constexpr f32 INF = INF; // 2e18 template <> constexpr f64 INF = INF; // 2e18 template <> constexpr f128 INF = INF; // 2e18 // input template std::istream& operator>>(std::istream& is, std::vector& v) { for (auto&& i : v) is >> i; return is; } template void in(T&... a) { (std::cin >> ... >> a); } void scan() {} template void scan(Head& head, Tail&... tail) { in(head); scan(tail...); } // definition & input #define INT(...) \ int __VA_ARGS__; \ scan(__VA_ARGS__) #define I64(...) \ i64 __VA_ARGS__; \ scan(__VA_ARGS__) #define U32(...) \ u32 __VA_ARGS__; \ scan(__VA_ARGS__) #define U64(...) \ u64 __VA_ARGS__; \ scan(__VA_ARGS__) #define F32(...) \ f32 __VA_ARGS__; \ scan(__VA_ARGS__) #define F64(...) \ f64 __VA_ARGS__; \ scan(__VA_ARGS__) #define F128(...) \ f128 __VA_ARGS__; \ scan(__VA_ARGS__) #define STR(...) \ std::string __VA_ARGS__; \ scan(__VA_ARGS__) #define CHR(...) \ char __VA_ARGS__; \ scan(__VA_ARGS__) #define VEC(type, name, size) \ std::vector name(size); \ scan(name) #define VEC2(type, name1, name2, size) \ std::vector name1(size), name2(size); \ for (int i = 0; i < size; i++) scan(name1[i], name2[i]) #define VEC3(type, name1, name2, name3, size) \ std::vector name1(size), name2(size), name3(size); \ for (int i = 0; i < size; i++) scan(name1[i], name2[i], name3[i]) #define VEC4(type, name1, name2, name3, name4, size) \ std::vector name1(size), name2(size), name3(size), name4(size); \ for (int i = 0; i < size; i++) scan(name1[i], name2[i], name3[i], name4[i]) #define VV(type, name, h, w) \ std::vector name((h), std::vector((w))); \ scan(name) // output template std::ostream& operator<<(std::ostream& os, const std::vector& v) { auto n = v.size(); for (size_t i = 0; i < n; i++) { if (i) os << ' '; os << v[i]; } return os; } template void out(const T&... a) { (std::cout << ... << a); } void print() { out('\n'); // std::cout.flush(); } template void print(Head&& head, Tail&&... tail) { out(head); if (sizeof...(Tail)) out(' '); print(tail...); } // for interactive problems void printflush() { out('\n'); std::cout.flush(); } template void printflush(Head&& head, Tail&&... tail) { out(head); if (sizeof...(Tail)) out(' '); printflush(tail...); } // bool output void YES(bool t = 1) { print(t ? "YES" : "NO"); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void NO(bool t = 1) { YES(!t); } void No(bool t = 1) { Yes(!t); } void no(bool t = 1) { yes(!t); } void POSSIBLE(bool t = 1) { print(t ? "POSSIBLE" : "IMPOSSIBLE"); } void Possible(bool t = 1) { print(t ? "Possible" : "Impossible"); } void possible(bool t = 1) { print(t ? "possible" : "impossible"); } void IMPOSSIBLE(bool t = 1) { POSSIBLE(!t); } void Impossible(bool t = 1) { Possible(!t); } void impossible(bool t = 1) { possible(!t); } void FIRST(bool t = 1) { print(t ? "FIRST" : "SECOND"); } void First(bool t = 1) { print(t ? "First" : "Second"); } void first(bool t = 1) { print(t ? "first" : "second"); } void SECOND(bool t = 1) { FIRST(!t); } void Second(bool t = 1) { First(!t); } void second(bool t = 1) { first(!t); } // I/O speed up struct SetUpIO { SetUpIO() { std::ios::sync_with_stdio(false); std::cin.tie(0); std::cout << std::fixed << std::setprecision(15); } } set_up_io; // find (x, y) s.t. ax + by = gcd(a, b) // a, b >= 0 // return {x, y, gcd(a, b)} template std::tuple extended_gcd(T a, T b) { if (b == 0) return {1, 0, a}; auto [y, x, g] = extended_gcd(b, a % b); return {x, y - (a / b) * x, g}; } // Reference: https://ja.wikipedia.org/wiki/ベズーの等式 // ax + by = c を解く (a, b >= 0) // return {x, y, gcd(a, b), has_solution} // 解が存在するとき // (1) a = 0, b = 0, c = 0 のとき // x, y は任意 // {0, 0, gcd(a, b) = 0, 1} を返す // (2) a = 0, c が b の倍数のとき // x は任意, y = c / b // {0, c / b, gcd(a, b) = b, 1} を返す // (3) b = 0, c が a の倍数のとき // y は任意, x = c / b // {c / a, 0, gcd(a, b) = a, 1} を返す // (4) a > 0, b > 0, c % gcd(a, b) = 0 のとき // x = x' + k * (b / gcd(a, b)), y = y' - k * (a / gcd(a, b)) // {x', y', gcd(a, b), 1} を返す // 解が存在しないとき // {-1, -1, -1, 0} を返す template std::tuple linear_diophantine(T a, T b, T c) { assert(a >= 0 and b >= 0); if (a == 0 and b == 0) { if (c == 0) return {0, 0, 0, 1}; return {-1, -1, -1, 0}; } if (a == 0) { // by = c if (c % b == 0) return {0, c / b, b, 1}; return {-1, -1, -1, 0}; } if (b == 0) { // ax = c if (c % a == 0) return {c / a, 0, a, 1}; return {-1, -1, -1, 0}; } // as + bt = gcd(a, b) から ax + by = c を求める // x = s * (c / gcd(a, b)), y = t * (c / gcd(a, b)) よりも x, y が小さくなる? // c = c' + a * dx + b * dy となるように c' を求める // (a, b は gcd(a, b) の倍数なので c' は gcd(a, b) の倍数) // x = dx + s * (c' / gcd(a, b)), y = dy + t * (c' / gcd(a, b)) が解となる // ax + by = a * dx + b * dy + (as + bt) * (c' / gcd(a, b)) = a * dx + b * dy + c' = c auto [s, t, g] = extended_gcd(a, b); if (c % g != 0) return {-1, -1, -1, 0}; T dx = c / a; c -= dx * a; T dy = c / b; c -= dy * b; T x = dx + s * (c / g); T y = dy + t * (c / g); return {x, y, g, 1}; } // 線形合同式 ax = b (mod m) を解く (m > 0) // 解が存在する場合 x = x' (mod h) となるときの最小の x' と h を返す // 解が存在しない場合 {-1, -1} を返す template std::pair linear_congruence(T a, T b, T m) { assert(m > 0); a = (a % m + m) % m; b = (b % m + m) % m; // ax = b (mod m) <=> ax + my = b となる (x, y) が存在 auto [x, y, g, is_ok] = linear_diophantine(a, m, b); if (!is_ok) return {-1, -1}; T h = m / g; x = (x % h + h) % h; return {x, h}; } // (rem, mod) std::pair chinese_remainder_theorem(const std::vector& r, const std::vector& m) { assert(r.size() == m.size()); const int n = (int)(r.size()); long long r0 = 0, m0 = 1; for (int i = 0; i < n; i++) { assert(m[i] >= 1); long long r1 = r[i] % m[i], m1 = m[i]; if (r1 < 0) r1 += m[i]; if (m0 < m1) { std::swap(r0, r1); std::swap(m0, m1); } // m0 > m1 if (m0 % m1 == 0) { if (r0 % m1 != r1) return {0, 0}; continue; } // (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1)) // r2 % m0 = r0 // -> r2 = r0 + x * m0 // r2 % m1 = r1 // -> (r0 + x * m0) % m1 = r1 // -> x * m0 = r1 - r0 (mod m1) auto [x, h] = linear_congruence(m0, r1 - r0, m1); if (x == -1 and h == -1) return {0, 0}; r0 += x * m0; m0 *= h; r0 %= m0; if (r0 < 0) r0 += m0; } return {r0, m0}; } using namespace std; void solve() { VEC2(i64, X, Y, 3); auto [r, m] = chinese_remainder_theorem(X, Y); show(r); show(m); if (m == 0) { print(-1); } else if (r == 0) { print(m); } else { print(r); } return; } int main() { solve(); return 0; }