#include #define ll long long #define INF 1000000005 #define MOD 1000000007 #define EPS 1e-10 #define rep(i,n) for(int i=0;i<(int)(n);++i) #define rrep(i,n) for(int i=(int)(n)-1;i>=0;--i) #define srep(i,s,t) for(int i=(int)(s);i<(int)(t);++i) #define each(a,b) for(auto& (a): (b)) #define all(v) (v).begin(),(v).end() #define len(v) (int)(v).size() #define zip(v) sort(all(v)),v.erase(unique(all(v)),v.end()) #define cmx(x,y) x=max(x,y) #define cmn(x,y) x=min(x,y) #define fi first #define se second #define pb push_back #define show(x) cout<<#x<<" = "<<(x)<auto&operator<<(ostream&o,pairp){return o<<"{"<auto&operator<<(ostream&o,sets){for(auto&e:s)o< auto&operator<<(ostream&o,priority_queueq){while(!q.empty())o<auto&operator<<(ostream&o,map&m){for(auto&e:m)o<auto&operator<<(ostream&o,vectorv){for(auto&e:v)o<void ashow(T t,A...a){cout< struct TRI{S fi;T se;U th;TRI(){}TRI(S f,T s,U t):fi(f),se(s),th(t){} bool operator<(const TRI&_)const{return(fi==_.fi)?((se==_.se)?(th<_.th):(se<_.se)):(fi<_.fi);}}; template auto&operator<<(ostream&o,TRI&t){return o<<"{"< P; typedef pair pll; typedef TRI tri; typedef vector vi; typedef vector vl; typedef vector vvi; typedef vector vvl; typedef vector

vp; typedef vector vd; typedef vector vs; const int MAX_N = 100005; class MPI : public vector { private: static constexpr int root = 3; static constexpr int MOD_ = 998244353; // 2^23 * 119 + 1 (2^22 桁以下の数の積を計算可能) static void trim_sign(MPI& num){ if(num.isZero()) num.sign = false; } static void trim_digit(MPI& num){ while(num.back() == 0 && (int)num.size() >= 2) num.pop_back(); } static bool abs_less(const MPI& a, const MPI& b){ if(a.size() != b.size()) return a.size() < b.size(); for(int index = (int)a.size() - 1; index >= 0; index--){ if(a[index] != b[index]) return a[index] < b[index]; } return false; } static void num_sbst(MPI& a, const MPI& b){ int n = (int)b.size(); a.resize(n); for(int i = 0; i < n; i++) a[i] = b[i]; } static void add(const MPI& a, const MPI& b, MPI& res){ num_sbst(res, a); int mx = (int)max(a.size(), b.size()); res.resize(mx, 0); int carry = 0; for(int i = 0; i < mx; i++){ int val = res[i] + ((i < (int)b.size()) ? b[i] : 0) + carry; carry = val/10; res[i] = val%10; } if(carry) res.push_back(1); } static void sub(const MPI& a, const MPI& b, MPI& res){ num_sbst(res, a); int carry = 0; for(int i = 0; i < (int)res.size(); i++){ int val = res[i] - carry - ((i < (int)b.size()) ? b[i] : 0); if(val < 0){ carry = 1, val += 10; }else{ carry = 0; } res[i] = val; } trim_digit(res), trim_sign(res); } static int add_(const int x, const int y) { return (x + y < MOD_) ? x + y : x + y - MOD_; } static int mul_(const int x, const int y) { return (ll)x * y % MOD_; } static int power(int x, int n){ int res = 1; while(n > 0){ if(n & 1) res = mul_(res, x); x = mul_(x, x); n >>= 1; } return res; } static int inverse(const int x) { return power(x, MOD_ - 2); } static void ntt(vector& a, const bool rev = false){ int i,j,k,s,t,v,w,wn; const int size = (int)a.size(); const int height = (int)log2(2 * size - 1); for(i = 0; i < size; i++){ j = 0; for(k = 0; k < height; k++) j |= (i >> k & 1) << (height - 1 - k); if(i < j) std::swap(a[i], a[j]); } for(i = 1; i < size; i *= 2) { w = power(root, (MOD_ - 1) / (i * 2)); if(rev) w = inverse(w); for(j = 0; j < size; j += i * 2){ wn = 1; for(k = 0; k < i; k++){ s = a[j + k], t = mul_(a[j + k + i], wn); a[j + k] = add_(s, t); a[j + k + i] = add_(s, MOD_ - t); wn = mul_(wn, w); } } } if(rev){ v = inverse(size); for (i = 0; i < size; i++) a[i] = mul_(a[i], v); } } static void mul(const MPI& a, const MPI& b, MPI& res){ const int size = (int)a.size() + (int)b.size() - 1; int t = 1; while (t < size) t <<= 1; vector A(t, 0), B(t, 0); for(int i = 0; i < (int)a.size(); i++) A[i] = a[i]; for(int i = 0; i < (int)b.size(); i++) B[i] = b[i]; ntt(A), ntt(B); for(int i = 0; i < t; i++) A[i] = mul_(A[i], B[i]); ntt(A, true); res.resize(size); int carry = 0; for(int i = 0; i < size; i++){ int val = A[i] + carry; carry = val / 10; res[i] = val % 10; } if(carry) res.push_back(carry); trim_digit(res), trim_sign(res); } bool isZero() const { return (int)size() == 1 && (*this)[0] == 0; } static bool div_geq(const MPI& mod, const MPI& num){ if((int)mod.size() != (int)num.size()) return (int)mod.size() > (int)num.size(); int n = (int)mod.size(); for(int i = 0; i < n; i++){ if(mod[i] != num[n-1-i]){ return mod[i] > num[n-1-i]; } } return true; } static void div_(const MPI& a, const MPI& b, MPI& quo, MPI& mod){ vector mult(9); mult[0] = b; for(int i = 0; i < 8; i++) mult[i+1] = mult[i] + b; for(int i = (int)a.size() - 1; i >= 0; i--){ if(mod.isZero()){ mod.back() = a[i]; }else{ mod.push_back(a[i]); } if(div_geq(mod, b)){ int l = 0, r = 9; reverse(mod.begin(), mod.end()); while(r-l>1){ int mid = (l+r)/2; if(mult[mid] > mod){ r = mid; }else{ l = mid; } } mod -= mult[l]; reverse(mod.begin(), mod.end()); quo.push_back(l+1); }else{ quo.push_back(0); } } reverse(quo.begin(), quo.end()); trim_digit(quo); reverse(mod.begin(), mod.end()); } public: ll to_ll() const { ll res = 0, dig = 1; for(int i = 0; i < (int)size(); i++){ res += dig * (*this)[i], dig *= 10; } if(sign) res = -res; return res; } string to_string() const { int n = (int)size() + sign; string s(n, ' '); if(sign) s[0] = '-'; for(int i = sign; i < n; i++) s[i] = (char)('0'+(*this)[n-1-i]); return s; } friend istream& operator>>(istream& is, MPI& num) { string s; is >> s; num = MPI(s); return is; } friend ostream& operator<<(ostream& os, const MPI& num) { if(num.sign) os << '-'; for(int i = (int)num.size()-1; i >= 0; i--) os << (char)('0'+num[i]); return os; } MPI& operator=(ll val) { *this = MPI(val); return *this; } bool operator<(const MPI& another) const { if(sign ^ another.sign) return sign; if(size() != another.size()) return (size() < another.size()) ^ sign; for(int index = (int)size() - 1; index >= 0; index--){ if((*this)[index] != another[index]) return ((*this)[index] < another[index]) ^ sign; } return false; } bool operator<(const ll num) const { return *this < MPI(num); } friend bool operator<(const ll num, const MPI& another){ return MPI(num) < another; } bool operator>(const MPI& another) const { return another < *this; } bool operator>(const ll num) const { return *this > MPI(num); } friend bool operator>(const ll num, const MPI& another){ return MPI(num) > another; } bool operator<=(const MPI& another) const { return !(*this > another); } bool operator<=(const ll num) const { return *this <= MPI(num); } friend bool operator<=(const ll num, const MPI& another){ return MPI(num) <= another; } bool operator>=(const MPI& another) const { return !(*this < another); } bool operator>=(const ll num) const { return *this >= MPI(num); } friend bool operator>=(const ll num, const MPI& another){ return MPI(num) >= another; } bool operator==(const MPI& another) const { if(sign ^ another.sign) return false; if(size() != another.size()) return false; for(int index = (int)size() - 1; index >= 0; index--){ if((*this)[index] != another[index]) return false; } return true; } bool operator==(const ll num) const { return *this == MPI(num); } friend bool operator==(const ll num, const MPI& another){ return MPI(num) == another; } bool operator!=(const MPI& another) const { return !(*this == another); } bool operator!=(const ll num) const { return *this != MPI(num); } friend bool operator!=(const ll num, const MPI& another){ return MPI(num) != another; } explicit operator bool() const noexcept { return !isZero(); } bool operator!() const noexcept { return !static_cast(*this); } explicit operator int() const noexcept { return (int)this->to_ll(); } explicit operator long long() const noexcept { return this->to_ll(); } MPI operator+() const { return *this; } MPI operator-() const { MPI res = *this; res.sign = !sign; return res; } friend MPI abs(const MPI& num){ MPI res = num; res.sign = false; return res; } MPI operator+(const MPI& num) const { MPI res; res.sign = sign; if(sign != num.sign){ if(abs_less(*this, num)){ res.sign = num.sign; sub(num, *this, res); return res; }else{ sub(*this, num, res); return res; } } add(*this, num, res); return res; } MPI operator+(ll num) const { return *this + MPI(num); } friend MPI operator+(ll a, const MPI& b){ return b + a; } MPI& operator+=(const MPI& num){ *this = *this + num; return *this; } MPI& operator+=(ll num){ *this = *this + num; return *this; } MPI& operator++(){ return *this += 1; } MPI operator++(int){ MPI res = *this; *this += 1; return res; } MPI operator-(const MPI& num) const { MPI res; res.sign = sign; if(sign != num.sign){ add(*this, num, res); return res; } if(abs_less(*this, num)){ res.sign = !sign; sub(num, *this, res); }else{ sub(*this, num, res); } return res; } MPI operator-(ll num) const { return *this - MPI(num); } friend MPI operator-(ll a, const MPI& b){ return MPI(a) - b; } MPI& operator-=(const MPI& num){ *this = *this - num; return *this; } MPI& operator-=(ll num){ *this = *this - num; return *this; } MPI& operator--(){ return *this -= 1; } MPI operator--(int){ MPI res = *this; *this -= 1; return res; } MPI operator*(const MPI& num) const { MPI res; res.sign = (sign^num.sign); mul(*this, num, res); return res; } MPI operator*(ll num) const { return *this * MPI(num); } friend MPI operator*(ll a, const MPI& b){ return b * a; } MPI& operator*=(const MPI& num){ *this = *this * num; return *this; } MPI& operator*=(ll num){ *this = *this * num; return *this; } MPI operator/(const MPI& num) const { MPI num_ = abs(num); MPI a, b; div_(*this, num_, a, b); a.sign = (sign^num.sign); trim_sign(a); return a; } MPI operator/(ll num) const { return *this / MPI(num); } friend MPI operator/(ll a, const MPI& b){ return MPI(a) / b; } MPI& operator/=(const MPI& num){ *this = *this / num; return *this; } MPI& operator/=(ll num){ *this = *this / num; return *this; } MPI operator%(const MPI& num) const { MPI num_ = abs(num); MPI a, b; div_(*this, num_, a, b); b.sign = sign; trim_sign(b); return b; } MPI operator%(ll num) const { return *this % MPI(num); } friend MPI operator%(ll a, const MPI& b){ return MPI(a) % b; } MPI& operator%=(const MPI& num){ *this = *this % num; return *this; } MPI& operator%=(ll num){ *this = *this % num; return *this; } MPI div2() const { MPI res; res.sign = sign; int n = (int)size(), carry = 0; for(int i = n-1; i >= 0; i--){ int val = (*this)[i]+carry*10; carry = val%2; if(i != n-1 || val >= 2) res.push_back(val/2); } reverse(res.begin(), res.end()); trim_digit(res); return res; } friend MPI sqrt(const MPI& x){ if(x <= MPI(0)) return MPI(0); MPI s = 1, t = x; while(s < t){ s = s + s, t = t.div2(); } do{ t = s, s = (x / s + s).div2(); }while(s < t); return t; } friend MPI log10(const MPI& x){ assert(x > MPI(0)); return MPI((int)x.size()); } friend MPI pow(MPI a, MPI b){ assert(b >= 0); MPI res(1); while(b){ if(b[0] % 2){ res *= a; } a *= a; b = b.div2(); } return res; } friend MPI mod_pow(MPI a, MPI b, const MPI& mod){ assert(b >= 0); MPI res(1); while(b){ if(b[0] % 2){ res = res * a % mod; } a = a * a % mod; b = b.div2(); } return res; } bool sign; MPI() : sign(false){ push_back(0); } MPI(ll val) : sign(false){ if(val == 0){ push_back(0); }else{ if(val < 0) sign = true, val = -val; while(val){ push_back(val%10); val /= 10; } } } MPI(const string& s) : sign(false){ if(s.empty()){ push_back(0); return; } if(s[0] == '-') sign = true; for(int i = (int)s.size() - 1; i >= sign; i--) push_back(s[i]-'0'); } }; MPI gcd(MPI a, MPI b){ MPI tmp; while(b) tmp = a, a = b, b = tmp % b; return a; } vector prime; bool is_prime[MAX_N]; void sieve(int n){ for(int i=0;i<=n;i++){ is_prime[i] = true; } is_prime[0] = is_prime[1] = false; for(int i=2;i<=n;i++){ if(is_prime[i]){ prime.push_back(i); for(int j=2*i;j<=n;j+=i){ is_prime[j] = false; } } } } int main() { MPI n; cin >> n; sieve(100); int id = 0; MPI a, res, hoge, p, q; while(true){ a = prime[id++]; if(n % a == 0){ cout << "! " << a << " " << n / a << endl; break; } cout << "? " << a << endl; cin >> res; if(res[0] % 2) continue; hoge = mod_pow(a, res.div2(), n) + 1; if(hoge == n) continue; p = gcd(n, hoge); cout << "! " << p << " " << n / p << endl; break; } return 0; }