#line 1 "main.cpp" #define PROBLEM "https://yukicoder.me/problems/no/187" #include #include #include #include #include #line 1 "/home/anqooqie/.proconlib/lib/ac-library/atcoder/modint.hpp" #include #include #include #ifdef _MSC_VER #include #endif #line 1 "/home/anqooqie/.proconlib/lib/ac-library/atcoder/internal_math.hpp" #line 5 "/home/anqooqie/.proconlib/lib/ac-library/atcoder/internal_math.hpp" #ifdef _MSC_VER #include #endif namespace atcoder { namespace internal { // @param m `1 <= m` // @return x mod m constexpr long long safe_mod(long long x, long long m) { x %= m; if (x < 0) x += m; return x; } // Fast modular multiplication by barrett reduction // Reference: https://en.wikipedia.org/wiki/Barrett_reduction // NOTE: reconsider after Ice Lake struct barrett { unsigned int _m; unsigned long long im; // @param m `1 <= m < 2^31` barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {} // @return m unsigned int umod() const { return _m; } // @param a `0 <= a < m` // @param b `0 <= b < m` // @return `a * b % m` unsigned int mul(unsigned int a, unsigned int b) const { // [1] m = 1 // a = b = im = 0, so okay // [2] m >= 2 // im = ceil(2^64 / m) // -> im * m = 2^64 + r (0 <= r < m) // let z = a*b = c*m + d (0 <= c, d < m) // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2 // ((ab * im) >> 64) == c or c + 1 unsigned long long z = a; z *= b; #ifdef _MSC_VER unsigned long long x; _umul128(z, im, &x); #else unsigned long long x = (unsigned long long)(((unsigned __int128)(z)*im) >> 64); #endif unsigned int v = (unsigned int)(z - x * _m); if (_m <= v) v += _m; return v; } }; // @param n `0 <= n` // @param m `1 <= m` // @return `(x ** n) % m` constexpr long long pow_mod_constexpr(long long x, long long n, int m) { if (m == 1) return 0; unsigned int _m = (unsigned int)(m); unsigned long long r = 1; unsigned long long y = safe_mod(x, m); while (n) { if (n & 1) r = (r * y) % _m; y = (y * y) % _m; n >>= 1; } return r; } // Reference: // M. Forisek and J. Jancina, // Fast Primality Testing for Integers That Fit into a Machine Word // @param n `0 <= n` constexpr bool is_prime_constexpr(int n) { if (n <= 1) return false; if (n == 2 || n == 7 || n == 61) return true; if (n % 2 == 0) return false; long long d = n - 1; while (d % 2 == 0) d /= 2; constexpr long long bases[3] = {2, 7, 61}; for (long long a : bases) { long long t = d; long long y = pow_mod_constexpr(a, t, n); while (t != n - 1 && y != 1 && y != n - 1) { y = y * y % n; t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } template constexpr bool is_prime = is_prime_constexpr(n); // @param b `1 <= b` // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g constexpr std::pair inv_gcd(long long a, long long b) { a = safe_mod(a, b); if (a == 0) return {b, 0}; // Contracts: // [1] s - m0 * a = 0 (mod b) // [2] t - m1 * a = 0 (mod b) // [3] s * |m1| + t * |m0| <= b long long s = b, t = a; long long m0 = 0, m1 = 1; while (t) { long long u = s / t; s -= t * u; m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b // [3]: // (s - t * u) * |m1| + t * |m0 - m1 * u| // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u) // = s * |m1| + t * |m0| <= b auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } // by [3]: |m0| <= b/g // by g != b: |m0| < b/g if (m0 < 0) m0 += b / s; return {s, m0}; } // Compile time primitive root // @param m must be prime // @return primitive root (and minimum in now) constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; int divs[20] = {}; divs[0] = 2; int cnt = 1; int x = (m - 1) / 2; while (x % 2 == 0) x /= 2; for (int i = 3; (long long)(i)*i <= x; i += 2) { if (x % i == 0) { divs[cnt++] = i; while (x % i == 0) { x /= i; } } } if (x > 1) { divs[cnt++] = x; } for (int g = 2;; g++) { bool ok = true; for (int i = 0; i < cnt; i++) { if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) { ok = false; break; } } if (ok) return g; } } template constexpr int primitive_root = primitive_root_constexpr(m); } // namespace internal } // namespace atcoder #line 1 "/home/anqooqie/.proconlib/lib/ac-library/atcoder/internal_type_traits.hpp" #line 7 "/home/anqooqie/.proconlib/lib/ac-library/atcoder/internal_type_traits.hpp" namespace atcoder { namespace internal { #ifndef _MSC_VER template using is_signed_int128 = typename std::conditional::value || std::is_same::value, std::true_type, std::false_type>::type; template using is_unsigned_int128 = typename std::conditional::value || std::is_same::value, std::true_type, std::false_type>::type; template using make_unsigned_int128 = typename std::conditional::value, __uint128_t, unsigned __int128>; template using is_integral = typename std::conditional::value || is_signed_int128::value || is_unsigned_int128::value, std::true_type, std::false_type>::type; template using is_signed_int = typename std::conditional<(is_integral::value && std::is_signed::value) || is_signed_int128::value, std::true_type, std::false_type>::type; template using is_unsigned_int = typename std::conditional<(is_integral::value && std::is_unsigned::value) || is_unsigned_int128::value, std::true_type, std::false_type>::type; template using to_unsigned = typename std::conditional< is_signed_int128::value, make_unsigned_int128, typename std::conditional::value, std::make_unsigned, std::common_type>::type>::type; #else template using is_integral = typename std::is_integral; template using is_signed_int = typename std::conditional::value && std::is_signed::value, std::true_type, std::false_type>::type; template using is_unsigned_int = typename std::conditional::value && std::is_unsigned::value, std::true_type, std::false_type>::type; template using to_unsigned = typename std::conditional::value, std::make_unsigned, std::common_type>::type; #endif template using is_signed_int_t = std::enable_if_t::value>; template using is_unsigned_int_t = std::enable_if_t::value>; template using to_unsigned_t = typename to_unsigned::type; } // namespace internal } // namespace atcoder #line 14 "/home/anqooqie/.proconlib/lib/ac-library/atcoder/modint.hpp" namespace atcoder { namespace internal { struct modint_base {}; struct static_modint_base : modint_base {}; template using is_modint = std::is_base_of; template using is_modint_t = std::enable_if_t::value>; } // namespace internal template * = nullptr> struct static_modint : internal::static_modint_base { using mint = static_modint; public: static constexpr int mod() { return m; } static mint raw(int v) { mint x; x._v = v; return x; } static_modint() : _v(0) {} template * = nullptr> static_modint(T v) { long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template * = nullptr> static_modint(T v) { _v = (unsigned int)(v % umod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } mint& operator*=(const mint& rhs) { unsigned long long z = _v; z *= rhs._v; _v = (unsigned int)(z % umod()); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { if (prime) { assert(_v); return pow(umod() - 2); } else { auto eg = internal::inv_gcd(_v, m); assert(eg.first == 1); return eg.second; } } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } private: unsigned int _v; static constexpr unsigned int umod() { return m; } static constexpr bool prime = internal::is_prime; }; template struct dynamic_modint : internal::modint_base { using mint = dynamic_modint; public: static int mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = internal::barrett(m); } static mint raw(int v) { mint x; x._v = v; return x; } dynamic_modint() : _v(0) {} template * = nullptr> dynamic_modint(T v) { long long x = (long long)(v % (long long)(mod())); if (x < 0) x += mod(); _v = (unsigned int)(x); } template * = nullptr> dynamic_modint(T v) { _v = (unsigned int)(v % mod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v += mod() - rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator*=(const mint& rhs) { _v = bt.mul(_v, rhs._v); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { auto eg = internal::inv_gcd(_v, mod()); assert(eg.first == 1); return eg.second; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } private: unsigned int _v; static internal::barrett bt; static unsigned int umod() { return bt.umod(); } }; template internal::barrett dynamic_modint::bt = 998244353; using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; using modint = dynamic_modint<-1>; namespace internal { template using is_static_modint = std::is_base_of; template using is_static_modint_t = std::enable_if_t::value>; template struct is_dynamic_modint : public std::false_type {}; template struct is_dynamic_modint> : public std::true_type {}; template using is_dynamic_modint_t = std::enable_if_t::value>; } // namespace internal } // namespace atcoder #line 1 "/home/anqooqie/.proconlib/tools/garner.hpp" #include #line 7 "/home/anqooqie/.proconlib/tools/garner.hpp" #include #line 9 "/home/anqooqie/.proconlib/tools/garner.hpp" #include #line 1 "/home/anqooqie/.proconlib/tools/mod.hpp" #line 1 "/home/anqooqie/.proconlib/tools/quo.hpp" #line 5 "/home/anqooqie/.proconlib/tools/quo.hpp" namespace tools { template constexpr ::std::common_type_t quo(const M lhs, const N rhs) { if (lhs >= 0) { return lhs / rhs; } else { if (rhs >= 0) { return -((-lhs - 1 + rhs) / rhs); } else { return (-lhs - 1 + -rhs) / -rhs; } } } } #line 6 "/home/anqooqie/.proconlib/tools/mod.hpp" namespace tools { template constexpr ::std::common_type_t mod(const M lhs, const N rhs) { if constexpr (::std::is_unsigned_v && ::std::is_unsigned_v) { return lhs % rhs; } else { return lhs - ::tools::quo(lhs, rhs) * rhs; } } } #line 1 "/home/anqooqie/.proconlib/tools/inv_mod.hpp" #line 1 "/home/anqooqie/.proconlib/tools/extgcd.hpp" #include #line 6 "/home/anqooqie/.proconlib/tools/extgcd.hpp" namespace tools { template ::std::tuple extgcd(T prev_r, T r) { T prev_s = 1; T prev_t = 0; T s = 0; T t = 1; while (r != 0) { const T q = ::tools::quo(prev_r, r); const T next_r = prev_r - q * r; prev_r = r; r = next_r; const T next_s = prev_s - q * s; prev_s = s; s = next_s; const T next_t = prev_t - q * t; prev_t = t; t = next_t; } if (prev_r < T(0)) prev_r = -prev_r; return {prev_s, prev_t, prev_r}; } } #line 7 "/home/anqooqie/.proconlib/tools/inv_mod.hpp" namespace tools { template constexpr T2 inv_mod(const T1 x, const T2 m) { const auto [x0, y0, gcd] = ::tools::extgcd(x, m); assert(gcd == 1); return ::tools::mod(x0, m); } } #line 13 "/home/anqooqie/.proconlib/tools/garner.hpp" // Source: https://qiita.com/drken/items/ae02240cd1f8edfc86fd // License: unknown // Author: drken namespace tools { template ::std::optional<::std::pair<::std::int_fast64_t, ::std::int_fast64_t>> garner(const Iterator& begin, const Iterator& end, const ModType& mod) { ::std::vector<::std::int_fast64_t> b, m; for (auto it = begin; it != end; ++it) { b.push_back(it->first); m.push_back(it->second); } ::std::int_fast64_t lcm = 1; for (::std::size_t i = 0; i < b.size(); ++i) { for (::std::size_t j = 0; j < i; ++j) { ::std::int_fast64_t g = ::std::gcd(m[i], m[j]); if ((b[i] - b[j]) % g != 0) return ::std::nullopt; m[i] /= g; m[j] /= g; ::std::int_fast64_t gi = ::std::gcd(m[i], g), gj = g / gi; do { g = ::std::gcd(gi, gj); gi *= g, gj /= g; } while (g != 1); m[i] *= gi, m[j] *= gj; b[i] %= m[i], b[j] %= m[j]; } } for (::std::size_t i = 0; i < b.size(); ++i) { (lcm *= m[i]) %= mod; } m.push_back(mod); ::std::vector<::std::int_fast64_t> coeffs(m.size(), 1); ::std::vector<::std::int_fast64_t> constants(m.size(), 0); for (::std::size_t k = 0; k < b.size(); ++k) { ::std::int_fast64_t t = ::tools::mod((b[k] - constants[k]) * ::tools::inv_mod(coeffs[k], m[k]), m[k]); for (::std::size_t i = k + 1; i < m.size(); ++i) { (constants[i] += t * coeffs[i]) %= m[i]; (coeffs[i] *= m[k]) %= m[i]; } } return ::std::make_optional<::std::pair<::std::int_fast64_t, ::std::int_fast64_t>>(constants.back(), lcm); } template ::std::optional<::std::pair> garner(const Iterator& begin, const Iterator& end) { const auto result = ::tools::garner(begin, end, M::mod()); if (!result) return ::std::nullopt; return ::std::make_optional<::std::pair>(M::raw(result->first), M::raw(result->second)); } } #line 10 "main.cpp" using i64 = std::int_fast64_t; int main() { std::cin.tie(nullptr); std::ios_base::sync_with_stdio(false); i64 N; std::cin >> N; std::vector> system(N); for (i64 i = 0; i < N; ++i) { std::cin >> system[i].first >> system[i].second; } const auto answer = tools::garner(system.begin(), system.end()); if (!answer) { std::cout << -1 << '\n'; return 0; } if (std::all_of(system.begin(), system.end(), [](const std::pair& pair) { return pair.first == 0; })) { std::cout << answer->second.val() << '\n'; return 0; } std::cout << answer->first.val() << '\n'; return 0; }