#include // created [2020/02/11] 13:40:33 #pragma GCC diagnostic ignored "-Wsign-compare" #pragma GCC diagnostic ignored "-Wsign-conversion" using i32 = int32_t; using i64 = int64_t; using u32 = uint32_t; using u64 = uint64_t; using uint = unsigned int; using usize = std::size_t; using ll = long long; using ull = unsigned long long; using ld = long double; template using arr = T (&)[n]; template using c_arr = const T (&)[n]; template using max_heap = std::priority_queue; template using min_heap = std::priority_queue, std::greater>; template constexpr T popcount(const T u) { return u ? static_cast(__builtin_popcountll(static_cast(u))) : static_cast(0); } template constexpr T log2p1(const T u) { return u ? static_cast(64 - __builtin_clzll(static_cast(u))) : static_cast(0); } template constexpr T msbp1(const T u) { return log2p1(u); } template constexpr T lsbp1(const T u) { return __builtin_ffsll(u); } template constexpr T clog(const T u) { return u ? log2p1(u - 1) : static_cast(u); } template constexpr bool ispow2(const T u) { return u and (static_cast(u) & static_cast(u - 1)) == 0; } template constexpr T ceil2(const T u) { return static_cast(1) << clog(u); } template constexpr T floor2(const T u) { return u == 0 ? static_cast(0) : static_cast(1) << (log2p1(u) - 1); } template constexpr bool btest(const T mask, const usize ind) { return static_cast((static_cast(mask) >> ind) & static_cast(1)); } template void bset(T& mask, const usize ind) { mask |= (static_cast(1) << ind); } template void breset(T& mask, const usize ind) { mask &= ~(static_cast(1) << ind); } template void bflip(T& mask, const usize ind) { mask ^= (static_cast(1) << ind); } template void bset(T& mask, const usize ind, const bool b) { (b ? bset(mask, ind) : breset(mask, ind)); } template constexpr T bcut(const T mask, const usize ind) { return ind == 0 ? static_cast(0) : static_cast((static_cast(mask) << (64 - ind)) >> (64 - ind)); } template bool chmin(T& a, const T& b) { return (a > b ? a = b, true : false); } template bool chmax(T& a, const T& b) { return (a < b ? a = b, true : false); } constexpr unsigned int mod = 1000000007; template constexpr T inf_v = std::numeric_limits::max() / 4; template constexpr Real pi_v = Real{3.141592653589793238462643383279502884}; auto mfp = [](auto&& f) { return [=](auto&&... args) { return f(f, std::forward(args)...); }; }; template T in() { T v; return std::cin >> v, v; } template T in_v(typename std::enable_if<(i == n), c_arr>::type) { return in(); } template auto in_v(typename std::enable_if<(i < n), c_arr>::type& szs) { const usize s = (usize)szs[i]; std::vector(szs))> ans(s); for (usize j = 0; j < s; j++) { ans[j] = in_v(szs); } return ans; } template auto in_v(c_arr szs) { return in_v(szs); } template auto in_t() { return std::tuple...>{in()...}; } struct io_init { io_init() { std::cin.tie(nullptr), std::ios::sync_with_stdio(false); std::cout << std::fixed << std::setprecision(20); } void clear() { std::cin.tie(), std::ios::sync_with_stdio(true); } } io_setting; int out() { return 0; } template int out(const T& v) { return std::cout << v, 0; } template int out(const std::vector& v) { for (usize i = 0; i < v.size(); i++) { if (i > 0) { std::cout << ' '; } out(v[i]); } return 0; } template int out(const std::pair& v) { return out(v.first), std::cout << ' ', out(v.second), 0; } template int out(const T& v, const Args... args) { return out(v), std::cout << ' ', out(args...), 0; } template int outln(const Args... args) { return out(args...), std::cout << '\n', 0; } template int outel(const Args... args) { return out(args...), std::cout << std::endl, 0; } # define SHOW(...) static_cast(0) constexpr ull TEN(const usize n) { return n == 0 ? 1ULL : TEN(n - 1) * 10ULL; } template auto make_v(typename std::enable_if<(i == n), c_arr>::type, const T& v = T{}) { return v; } template auto make_v(typename std::enable_if<(i < n), c_arr>::type szs, const T& v = T{}) { const usize s = (usize)szs[i]; return std::vector(szs, v))>(s, make_v(szs, v)); } template auto make_v(c_arr szs, const T& t = T{}) { return make_v(szs, t); } template T gcd(const T& a, const T& b) { return a < 0 ? gcd(-a, b) : b < 0 ? gcd(a, -b) : (a > b ? gcd(b, a) : a == 0 ? b : gcd(b % a, a)); } template T lcm(const T& a, const T& b) { return a / gcd(a, b) * b; } template constexpr std::pair extgcd(const T a, const T b) { if (b == 0) { return std::pair{1, 0}; } const auto g = gcd(a, b), da = std::abs(b) / g; const auto p = extgcd(b, a % b); const auto x = (da + p.second % da) % da, y = (g - a * x) / b; return {x, y}; } template constexpr T inverse(const T a, const T mod) { return extgcd(a, mod).first; } template class modint_base { public: template static std::enable_if_t mod() { return mod_ref(); } template static constexpr std::enable_if_t mod() { return mod_value; } template static void set_mod(const std::enable_if_t mod) { mod_ref() = mod, inv_ref() = {1, 1}; } modint_base() : v{0} {} modint_base(const ll val) : v{norm(static_cast(val % static_cast(mod()) + static_cast(mod())))} {} modint_base(const modint_base& n) : v{n()} {} explicit operator bool() const { return v != 0; } bool operator!() const { return not static_cast(*this); } modint_base& operator=(const modint_base& m) { return v = m(), (*this); } modint_base& operator=(const ll val) { return v = norm(uint(val % static_cast(mod()) + static_cast(mod()))), (*this); } friend modint_base operator+(const modint_base& m) { return m; } friend modint_base operator-(const modint_base& m) { return make(norm(mod() - m.v)); } friend modint_base operator+(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + m2.v)); } friend modint_base operator-(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + mod() - m2.v)); } friend modint_base operator*(const modint_base& m1, const modint_base& m2) { return make(static_cast(static_cast(m1.v) * static_cast(m2.v) % static_cast(mod()))); } friend modint_base operator/(const modint_base& m1, const modint_base& m2) { return m1 * inv(m2.v); } friend modint_base operator+(const modint_base& m, const ll val) { return modint_base{static_cast(m.v) + val}; } friend modint_base operator-(const modint_base& m, const ll val) { return modint_base{static_cast(m.v) - val}; } friend modint_base operator*(const modint_base& m, const ll val) { return modint_base{static_cast(m.v) * (val % static_cast(mod()))}; } friend modint_base operator/(const modint_base& m, const ll val) { return modint_base{static_cast(m.v) * inv(val)}; } friend modint_base operator+(const ll val, const modint_base& m) { return modint_base{static_cast(m.v) + val}; } friend modint_base operator-(const ll val, const modint_base& m) { return modint_base{-static_cast(m.v) + val}; } friend modint_base operator*(const ll val, const modint_base& m) { return modint_base{static_cast(m.v) * (val % static_cast(mod()))}; } friend modint_base operator/(const ll val, const modint_base& m) { return modint_base{val * inv(static_cast(m.v))}; } friend modint_base& operator+=(modint_base& m1, const modint_base& m2) { return m1 = m1 + m2; } friend modint_base& operator-=(modint_base& m1, const modint_base& m2) { return m1 = m1 - m2; } friend modint_base& operator*=(modint_base& m1, const modint_base& m2) { return m1 = m1 * m2; } friend modint_base& operator/=(modint_base& m1, const modint_base& m2) { return m1 = m1 / m2; } friend modint_base& operator+=(modint_base& m, const ll val) { return m = m + val; } friend modint_base& operator-=(modint_base& m, const ll val) { return m = m - val; } friend modint_base& operator*=(modint_base& m, const ll val) { return m = m * val; } friend modint_base& operator/=(modint_base& m, const ll val) { return m = m / val; } friend modint_base operator^(const modint_base& m, const ll n) { return power(m.v, n); } friend modint_base& operator^=(modint_base& m, const ll n) { return m = m ^ n; } friend bool operator==(const modint_base& m1, const modint_base& m2) { return m1.v == m2.v; } friend bool operator!=(const modint_base& m1, const modint_base& m2) { return not(m1 == m2); } friend bool operator==(const modint_base& m, const ll val) { return m.v == norm(static_cast(static_cast(mod()) + val % static_cast(mod()))); } friend bool operator!=(const modint_base& m, const ll val) { return not(m == val); } friend bool operator==(const ll val, const modint_base& m) { return m.v == norm(static_cast(static_cast(mod()) + val % static_cast(mod()))); } friend bool operator!=(const ll val, const modint_base& m) { return not(m == val); } friend std::istream& operator>>(std::istream& is, modint_base& m) { ll v; return is >> v, m = v, is; } friend std::ostream& operator<<(std::ostream& os, const modint_base& m) { return os << m(); } uint operator()() const { return v; } static modint_base small_inv(const usize n) { auto& in = inv_ref(); if (n < in.size()) { return in[n]; } for (usize i = in.size(); i <= n; i++) { in.push_back(-in[modint_base::mod() % i] * (modint_base::mod() / i)); } return in.back(); } std::pair quad() const { const auto ans = quad_r(v, mod()); ll x = std::get<0>(ans), y = std::get<1>(ans); if (y < 0) { x = -x, y = -y; } return {x, y}; } private: static std::tuple quad_r(const ll r, const ll p) // r = x/y (mod p), (x,y,z) s.t. x=yr+pz { if (std::abs(r) <= 1000) { return {r, 1, 0}; } ll nr = p % r, q = p / r; if (nr * 2LL >= r) { nr -= r, q++; } if (nr * 2LL <= -r) { nr += r, q--; } const auto sub = quad_r(nr, r); const ll x = std::get<0>(sub), z = std::get<1>(sub), y = std::get<2>(sub); return {x, y - q * z, z}; } template static std::enable_if_t mod_ref() { static UInt mod = 0; return mod; } static uint norm(const uint x) { return x < mod() ? x : x - mod(); } static modint_base make(const uint x) { modint_base m; return m.v = x, m; } static modint_base power(modint_base x, ull n) { modint_base ans = 1; for (; n; n >>= 1, x *= x) { if (n & 1) { ans *= x; } } return ans; } static modint_base inv(const ll v) { return v <= 2000000 ? small_inv(static_cast(v)) : modint_base{inverse(v, static_cast(mod()))}; } static std::vector& inv_ref() { static std::vector in{1, 1}; return in; } uint v; }; template using modint = modint_base; template using dynamic_modint = modint_base; template class wavelet_matrix { public: using value_type = u64; template wavelet_matrix(const InIt first, const InIt last) : sz{static_cast(std::distance(first, last))} { for (usize i = 0; i < lg; i++) { zero[i] = 0, table[i] = fid(sz); } std::vector v{first, last}; std::vector z(sz), o(sz); usize zn = 0, on = 0; for (usize d = 0; d < lg; d++) { zn = 0, on = 0; for (usize i = 0; i < sz; i++) { const bool b = btest(v[i], lg - d - 1); table[d].set(i, b), (b ? o[on++] : z[zn++]) = v[i]; } table[d].build(), zero[d] = zn, std::swap(v, z); for (usize i = 0; i < on; i++) { v[zn + i] = o[i]; } } } usize less_than(usize l, usize r, const value_type v) const { assert(l <= r); usize ans = 0; for (usize i = 0; i < lg; i++) { const usize zl = table[i].zero(l), zr = table[i].zero(r), z = zero[i]; if (btest(v, lg - i - 1)) { ans += zr - zl, l += z - zl, r += z - zr; } else { l = zl, r = zr; } } return ans; } usize range_freq(const usize l, const usize r, const value_type vmin, const value_type vsup) const { return less_than(l, r, vsup) - less_than(l, r, vmin); } value_type quantile(usize l, usize r, usize n) const { assert(l <= r), assert(r - l > n); value_type ans = 0; for (usize i = 0; i < lg; i++) { const usize zl = table[i].zero(l), zr = table[i].zero(r), z = zero[i]; if (n >= zr - zl) { ans += value_type(1) << (lg - i - 1), l += z - zl, r += z - zr, n -= (zr - zl); } else { l = zl, r = zr; } } return ans; } private: static constexpr usize bucket_size = sizeof(value_type) * 8; static constexpr usize bslog = log2p1(bucket_size) - 1; static constexpr usize wind(const usize n) { return n >> (bslog); } static constexpr usize bind(const usize n) { return bcut(n, bslog); } class fid { private: usize sz, bn; std::vector data; std::vector large; public: fid() : sz{0} {} fid(const usize b) : sz{b}, bn{wind(sz) + 2}, data(bn, 0), large(bn, 0) {} void build() { for (usize i = 1; i < large.size(); i++) { large[i] = large[i - 1] + popcount(data[i - 1]); } } bool operator[](const usize n) const { return btest(data[wind(n)], bind(n)); } void set(const usize n, const bool b) { bset(data[wind(n)], bind(n), b); } usize one(const usize n) const { return large[wind(n)] + popcount(bcut(data[wind(n)], bind(n))); } usize zero(const usize n) const { return n - one(n); } }; const usize sz; std::array zero; std::array table; }; template class lazyseg { public: using monoid_act_type = MonoidAct; using value_monoid_type = typename monoid_act_type::value_monoid_type; using operator_monoid_type = typename monoid_act_type::operator_monoid_type; using value_type = typename value_monoid_type::value_type; using operator_type = typename operator_monoid_type::operator_type; lazyseg(const usize sz, const value_type initial = value_monoid_type::id()) : sz{sz}, depth{clog(sz)}, half{static_cast(1) << depth}, val(half << 1, value_monoid_type::id()), op(half << 1, operator_monoid_type::id()) { if (initial != value_monoid_type::id()) { std::fill(val.begin() + half, val.end(), initial); for (usize i = half - 1; i >= 1; i--) { up(i); } } } template lazyseg(const InIt first, const InIt last) : sz{static_cast(std::distance(first, last))}, depth{clog(sz)}, half{static_cast(1) << depth}, val(half << 1, value_monoid_type::id()), op(half << 1, operator_monoid_type::id()) { std::copy(first, last, val.begin() + half); for (usize i = half - 1; i >= 1; i--) { up(i); } } value_type get(const usize a) { return assert(a < sz), fold(a, a + 1); } void set(usize a, const value_type& v) { assert(a < sz); top_down(a += half), top_down(a + 1), op[a] = operator_monoid_type::id(), val[a] = v; while (a >>= 1) { up(a); } } value_type fold(usize l, usize r) { assert(l < r), assert(r <= sz); top_down(l += half), top_down(r += half); value_type accl = value_monoid_type::id(), accr = value_monoid_type::id(); for (; l < r; l >>= 1, r >>= 1) { if (l & 1) { accl = value_monoid_type::merge(accl, val[l++]); } if (r & 1) { accr = value_monoid_type::merge(val[--r], accr); } } return value_monoid_type::merge(accl, accr); } void act(usize l, usize r, const operator_type& f) { assert(l < r), assert(r <= sz); const usize lin = l + half, rin = r + half; top_down(l += half), top_down(r += half); for (usize ls = 1, rs = 1; l < r; l >>= 1, r >>= 1, ls <<= 1, rs <<= 1) { if (l & 1) { update(l++, f, ls); } if (r & 1) { update(--r, f, rs); } } bottom_up(lin), bottom_up(rin); } usize size() const { return sz; } friend std::ostream& operator<<(std::ostream& os, const lazyseg& lseg) { auto lseg2 = lseg; os << "["; for (usize i = 0; i < lseg.sz; i++) { os << lseg2.get(i) << (i + 1 == lseg2.sz ? "" : ","); } return (os << "]\n"); } private: void up(const usize i) { val[i] = value_monoid_type::merge(val[i << 1], val[i << 1 | 1]); } void update(const usize a, const operator_type& f, const usize l) { op[a] = operator_monoid_type::compose(f, op[a]), val[a] = monoid_act_type::apply(f, val[a], l); } void down(const usize a, const usize l) { update(a << 1, op[a], l >> 1), update(a << 1 | 1, op[a], l >> 1), op[a] = operator_monoid_type::id(); } void top_down(const usize a) { const usize b = (a / (a & -a)) >> 1; for (usize i = 0, l = half; i < depth; i++, l >>= 1) { const usize v = a >> (depth - i); if (v > b) { break; } down(v, l); } } void bottom_up(usize a) { a = (a / (a & -a)) >> 1; for (; a >= 1; a >>= 1) { up(a); } } const usize sz, depth, half; std::vector val; std::vector op; }; struct accum_sum // a_l+2*a_{l+1}+3*a_{l+2}+...+(r-l)*a_{r-1} { struct value_type { ll asum = 0; ll sum = 0; ll len = 0; friend std::ostream& operator<<(std::ostream& os, const value_type& v) { os << "[asum=" << v.asum << ",sum=" << v.sum << "]"; return os; } }; accum_sum() = delete; static value_type merge(const value_type& a, const value_type& b) { value_type ans; ans.len = a.len + b.len; ans.sum = a.sum + b.sum; ans.asum = a.asum + b.asum + b.sum * a.len; return ans; } static value_type id() { return value_type{}; } }; struct plus { using operator_type = ll; plus() = delete; static operator_type compose(const operator_type& a, const operator_type& b) { return a + b; } static constexpr operator_type id() { return operator_type{}; } }; struct mact { using value_monoid_type = accum_sum; using operator_monoid_type = plus; using value_type = typename value_monoid_type::value_type; using operator_type = typename operator_monoid_type::operator_type; mact() = delete; static value_type apply(const operator_type& f, const value_type& x, const usize) { value_type ans; ans.len = x.len; ans.sum = x.sum + f * x.len; ans.asum = x.asum + f * x.len * (x.len + 1) / 2; return ans; } }; int main() { const auto P = in(); using mint = dynamic_modint<0>; mint::set_mod(P); auto N = in(); if (N == 1) { return outln(0); } bool inv = false; if (N * N < P) { inv = true; N = (mint(1) / N)(); } std::vector bs; for (ll i = 0, q = 1; i < P; q++) { bs.push_back(i * N % P); i = (P * q - 1) / N + 1; } wavelet_matrix<33> wm(bs.begin(), bs.end()); const ll L = P / N + 1; // 0-N-2N-...-(L-1)N std::vector vs(L); for (int i = 0; i < L; i++) { vs[i] = {0, 0, 1}; } lazyseg seg(vs.begin(), vs.end()); ll ans = 0; for (int i = 0; i < bs.size(); i++) { SHOW(seg); const ll b = bs[i]; const ll l = (P - b - 1) / N + 1; SHOW(i, b, l); ll plus = seg.fold(0, L).asum + (l >= L ? 0LL : seg.fold(l, L).asum); SHOW(plus); plus -= wm.less_than(0, i, bs[i]) * l; ans += plus; seg.act(0, l, 1LL); } if (inv) { ans = P * (P - 1) / 2 - ans; } outln(ans % 2); return 0; }