#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 using namespace std; using lint = int64_t; using pint = pair; using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } // int floor_lg(__int128_t x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template pair operator+(const pair &l, const pair &r) { return make_pair(l.first + r.first, l.second + r.second); } template pair operator-(const pair &l, const pair &r) { return make_pair(l.first - r.first, l.second - r.second); } template vector sort_unique(vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template istream &operator>>(istream &is, vector &vec) { for (auto &v : vec) is >> v; return is; } template ostream &operator<<(ostream &os, const vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } #if __cplusplus >= 201703L template istream &operator>>(istream &is, tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template ostream &operator<<(ostream &os, const tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } #endif template ostream &operator<<(ostream &os, const deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const pair &pa) { os << '(' << pa.first << ',' << pa.second << ')'; return os; } template ostream &operator<<(ostream &os, const map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl #define dbgif(cond, x) ((cond) ? cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl : cerr) #else #define dbg(x) 0 #define dbgif(cond, x) 0 #endif // \sum_{i=0}^{n-1} floor((ai + b) / m) // 0 <= n < 2e32 // 1 <= m < 2e32 // 0 <= a, b < m // Complexity: O(lg(m)) __int128_t floor_sum(__int128_t n, __int128_t m, __int128_t a, __int128_t b) { auto safe_mod = [](__int128_t x, __int128_t m) -> __int128_t { x %= m; if (x < 0) x += m; return x; }; auto floor_sum_unsigned = [](__uint128_t n, __uint128_t m, __uint128_t a, __uint128_t b) -> __uint128_t { __uint128_t ans = 0; while (true) { if (a >= m) { ans += n * (n - 1) / 2 * (a / m); a %= m; } if (b >= m) { ans += n * (b / m); b %= m; } __uint128_t y_max = a * n + b; if (y_max < m) break; // y_max < m * (n + 1) // floor(y_max / m) <= n n = (__uint128_t)(y_max / m); b = (__uint128_t)(y_max % m); std::swap(m, a); } return ans; }; __uint128_t ans = 0; if (a < 0) { __uint128_t a2 = safe_mod(a, m); ans -= 1ULL * n * (n - 1) / 2 * ((a2 - a) / m); a = a2; } if (b < 0) { __uint128_t b2 = safe_mod(b, m); ans -= 1ULL * n * ((b2 - b) / m); b = b2; } return ans + floor_sum_unsigned(n, m, a, b); } int main() { lint P, Q; cin >> P >> Q; const __int128_t D = 1000000007LL * 1000000009; const lint cnt = floor_sum(Q, Q, P, 0); dbg(cnt); lint lo = 0, hi = D; // lo / D 以下にあるやつは足りない hi / D 以下にあるやつは足りる while (hi - lo > 1) { lint c = (lo + hi) / 2; (floor_sum(Q, D, c, 0) < cnt ? lo : hi) = c; } dbg(lo); // P / Q と lo / D で差分が出るところは? lint xsame = 0, xdiff = Q - 1; while (xdiff - xsame > 1) { const lint x = (xdiff + xsame) / 2; if (floor_sum(x + 1, Q, P, 0) - floor_sum(x + 1, D, lo, 0)) { xdiff = x; } else { xsame = x; } } lint ret_l_den = xdiff, ret_l_num = (P * xdiff - 1) / Q; lo = 0, hi = D; // lo / D 以下にあるやつは以下 hi / D 以下にあるやつは過剰 while (hi - lo > 1) { lint c = (lo + hi) / 2; (floor_sum(Q, D, c, 0) <= cnt ? lo : hi) = c; } dbg(lo); xsame = 0, xdiff = Q - 1; while (xdiff - xsame > 1) { const lint x = (xdiff + xsame) / 2; if (floor_sum(x + 1, Q, P, 0) - floor_sum(x + 1, D, hi, 0)) { xdiff = x; } else { xsame = x; } } lint ret_r_den = xdiff, ret_r_num = P * xdiff / Q + 1; dbg(make_tuple(ret_l_num, ret_l_den, ret_r_num, ret_r_den)); cout << ret_l_den + ret_l_num + ret_r_den + ret_r_num << endl; }