#include using namespace std; //入力が必ず-mod struct modint{ //mod変更が不可能. public: long long v = 0; static void setmod(int m){} //飾り. static constexpr long long getmod(){return mod;} modint(){v = 0;} modint(int a){v = a<0?a+mod:a;} modint(long long a){v = a<0?a+mod:a;} modint(unsigned int a){v = a;} modint(unsigned long long a){v = a;} long long val()const{return v;} modint &operator=(const modint &b) = default; modint operator+()const{return (*this);} modint operator-()const{return modint(0)-(*this);} modint operator+(const modint b)const{return modint(v)+=b;} modint operator-(const modint b)const{return modint(v)-=b;} modint operator*(const modint b)const{return modint(v)*=b;} modint operator/(const modint b)const{return modint(v)/=b;} modint operator+=(const modint b){ v += b.v; if(v >= mod) v -= mod; return *this; } modint operator-=(const modint b){ v -= b.v; if(v < 0) v += mod; return *this; } modint operator*=(const modint b){v = v*b.v%mod; return *this;} modint operator/=(modint b){ //b!=0 mod素数が必須. if(b == 0) assert(false); int left = mod-2; while(left){if(left&1) *this *= b; b *= b; left >>= 1;} return *this; } modint operator++(){*this += 1; return *this;} modint operator--(){*this -= 1; return *this;} modint operator++(int){*this += 1; return *this;} modint operator--(int){*this -= 1; return *this;} bool operator==(const modint b)const{return v == b.v;} bool operator!=(const modint b)const{return v != b.v;} bool operator>(const modint b)const{return v > b.v;} bool operator>=(const modint b)const{return v >= b.v;} bool operator<(const modint b)const{return v < b.v;} bool operator<=(const modint b)const{return v <= b.v;} modint pow(long long n)const{ modint ret = 1,p = v; if(n < 0) p = p.inv(),n = -n; while(n){ if(n&1) ret *= p; p *= p; n >>= 1; } return ret; } modint inv()const{return modint(1)/v;} //素数mod必須. }; using mint = modint<998244353>; namespace to_fold{ __int128_t safemod(__int128_t a,long long m){a %= m; if(a < 0) a += m; return a;} pair invgcd(long long a,long long b){ //return {gcd(a,b),x} (xa≡g(mod b)) a = safemod(a,b); if(a == 0) return {b,0}; long long x = 0,y = 1,memob = b; while(a){ long long q = b/a; b -= a*q; swap(x,y); y -= q*x; swap(a,b); } if(x < 0) x += memob/b; return {b,x}; } template long long Garner(const vector &A,const vector &M){ __int128_t mulM = 1,x = A.at(0)%M.at(0); //Mの要素のペア互いに素必須. for(int i=1; i struct fftinfo{ static bool First; static mint g,sum_e[30],sum_ie[30]; //sum_e[i]=Π[j=0~i-1]ies[j] * es[i],sum_ie[i]=Π[i=0~j-1]es[j] * ies[i]. static mint divpow2[30]; //div[i] = 1/(2^i). static mint Zeta[30]; fftinfo(){ if(!First) return; First = false; const long long mod = mint::getmod(); if(mod == 998244353) g = 3; else if(mod == 754974721) g = 11; else if(mod == 167772161) g = 3; else if(mod == 469762049) g = 3; else assert(false); //現状RE. mint es[30],ies[30]; //es[i]^(2^(2+i))=1. int cnt2 = countzero(mod-1); mint e = g.pow((mod-1)>>cnt2),ie = e.inv(); for(int i=cnt2; i>=2; i--){ //e^(2^i)=1; es[i-2] = e,e *= e; ies[i-2] = ie,ie *= ie; } mint rot = 1; for(int i=0; i<=cnt2-2; i++) sum_e[i] = es[i]*rot,rot *= ies[i]; rot = 1; for(int i=0; i<=cnt2-2; i++) sum_ie[i] = ies[i]*rot,rot *= es[i]; mint div2n = 1,div2 = mint(1)/2; for(int i=0; i<30; i++) divpow2[i] = div2n,div2n *= div2; for(int i=0; i<=cnt2; i++) Zeta[i] = g.pow((mod-1)/(2< bool fftinfo::First=true; template mint fftinfo::g; template mint fftinfo::sum_e[30]; template mint fftinfo::sum_ie[30]; template mint fftinfo::divpow2[30]; template mint fftinfo::Zeta[30]; template void NTT(vector &A){ //ACLを超参考にしてる. int n = A.size(); assert((n&-n) == n); fftinfo info; int h = countzero(n); for(int ph=1; ph<=h; ph++){ int w = 1<<(ph-1),p = 1<<(h-ph); mint rot = 1; for(int s=0; s void INTT(vector &A){ int n = A.size(); assert((n&-n) == n); fftinfo info; const unsigned int mod = mint::getmod(); int h = countzero(n); for(int ph=h; ph>0; ph--){ int w = 1<<(ph-1),p = 1<<(h-ph); mint irot = 1; for(int s=0; s vector convolution(vector A,vector B){ //mintじゃないのを突っ込まないように!!!. int siza = A.size(),sizb = B.size(),sizc = siza+sizb-1,N = 1; if(siza == 0 || sizb == 0) return {}; if(min(siza,sizb) <= 60){ //naive. vector ret(sizc); if(siza >= sizb){for(int i=0; i convolution_ll(const vector &A,const vector &B){ //long longに収まる範囲. int siza = A.size(),sizb = B.size(),sizc = siza+sizb-1; if(siza == 0 || sizb == 0) return {}; vector ret(sizc); if(min(siza,sizb) <= 200){ //naive 200はやばい?. vector ret(sizc); if(siza >= sizb){for(int i=0; i; using mint2 = modint; using mint3 = modint; vector a1(siza),b1(sizb); vector a2(siza),b2(sizb); vector a3(siza),b3(sizb); for(int i=0; i C1 = convolution(a1,b1); for(int i=0; i C2 = convolution(a2,b2); for(int i=0; i C3 = convolution(a3,b3); vector offset = {0,0,m1m2m3,2*m1m2m3,3*m1m2m3}; for(int i=0; i vector convolution_llmod(const vector &A,const vector &B){ int siza = A.size(),sizb = B.size(),sizc = siza+sizb-1; if(siza == 0 || sizb == 0) return {}; vector ret(sizc); if(min(siza,sizb) <= 200){ for(int i=0; i; using mint2 = modint; using mint3 = modint; vector a1(siza),b1(sizb); vector a2(siza),b2(sizb); vector a3(siza),b3(sizb); for(int i=0; i C1 = convolution(a1,b1); for(int i=0; i C2 = convolution(a2,b2); for(int i=0; i C3 = convolution(a3,b3); for(int i=0; i A = {C1.at(i).v,C2.at(i).v,C3.at(i).v}; vector M = {mod1,mod2,mod3}; ret.at(i) = Garner(A,M); } return ret; } vector convolution_int(const vector &A,const vector &B){ //intに収まる範囲. if(A.size() == 0 || B.size() == 0) return {}; vector ret; if(min(A.size(),B.size()) <= 60){ ret.resize(A.size()+B.size()-1); for(int i=0; i; vector X(A.size()),Y(B.size()),Z; for(int i=0; i void NTTdoubling(vector &A){ //NTTの原理を忘れているため何やってるのか意味が分からない NTT-friendly専用. //INTT->resize(2倍)->NTTの代わりにcopy->INTT->謎の操作->NTT->push sizeが小さい時は効率悪いらしいよ. int n = A.size(); fftinfo info; vector B = A; INTT(B); mint rot = 1,zeta = info.Zeta[countzero(n)]; for(auto &v : B) v *= rot,rot *= zeta; NTT(B); A.reserve(n<<1); for(auto &v : B) A.push_back(v); } bool isNTTfriendly(long long mod){ if(mod == 998244353 || mod == 754974721 || mod == 16777216 || mod == 469762049) return true; return false; //現状false 原子根求める機能を追加してから. int have2 = countzero(mod-1); return have2 >= 20;//とりあえず2^20でokとする; } } using namespace to_fold; struct bigint{ //__int128_t以上の数にも対応. //計算量 普通O(桁数),掛け算O(桁数log桁数),割り算O(桁数log桁数)(重い). public: bool isMinus; //負かどうか. vector V;//絶対値. //先頭の要素の方が小さい位 5108->{8,0,1,5}. private: void slide(){ for(int i=0; i|a|<|b|,1->|a|>|b|,2->|a|==|b|. if(V.size() < b.V.size()) return 0; if(V.size() > b.V.size()) return 1; for(int i=V.size(); i--;){ if(V.at(i) == b.V.at(i)) continue; return V.at(i)>b.V.at(i); } return 2; } public: bigint():isMinus(false),V({}){} template bigint(T a){ if(a < 0) isMinus = true,a = -a; else isMinus = false; V = {}; while(a) V.push_back(a%10),a /= 10; } bigint(const string &s){ V = {}; if(s.size() == 0) isMinus = false; else{ if(s.at(0) == '-') isMinus = true; else isMinus = false; for(int i=s.size(); i--;) V.push_back(s.at(i)-'0'); if(isMinus) V.pop_back(); while(V.size() && V.back() == '0') V.pop_back(); } } bigint(const bigint &n){isMinus = n.isMinus; V = n.V;} bigint &operator=(const bigint &b) = default; bigint operator+(const bigint &b){ bigint ret; if(isMinus == b.isMinus){ ret = *this; ret.V.resize(max(V.size(),b.V.size())); for(int i=0; i=0; i--){ int ok = 0; bigint now; for(int k=i; k= b) now -= b,ok++; else break; } now.V.resize(memo); for(int k=i; k one.V.size()) two.V.pop_back(); two.V.push_back(2); bigint next = invB*(two-one); invB.V.assign(2,0); invB.V.at(1) = next.V.back(); next.V.pop_back(); invB.V.at(0) = next.V.back(); } while(accu < need){ //精度倍々になる. bigint one = invB*b; two.V.pop_back(); while(two.V.size() < one.V.size()) two.V.push_back(0); two.V.push_back(2); bigint next = invB*(two-one); accu = min(accu<<1,need); invB.V.clear(); for(int i=max(0,(int)next.V.size()-accu); i= b){ //+1の可能性. int digit = 0; ret.V.push_back(0); ret.V.at(digit)++; while(ret.V.at(digit) == 10) ret.V.at(digit) = 0,ret.V.at(++digit)++; if(ret.V.back() == 0) ret.V.pop_back(); } return ret; } bigint operator%(const bigint &b){ //a>=0,b>0を仮定. bigint ret = *this; ret -= ret/b*b; return ret; } bigint operator+=(const bigint &b){return *this=*this+b;} bigint operator-=(const bigint &b){return *this=*this-b;} bigint operator*=(const bigint &b){return *this=*this*b;} bigint operator/=(const bigint &b){return *this=*this/b;} bigint operator%=(const bigint &b){return *this=*this%b;} bool operator==(const bigint &b){ if(V.size() != b.V.size() || isMinus != b.isMinus) return false; return V == b.V; } bool operator<(const bigint &b){ if(isMinus && !b.isMinus) return true; if(!isMinus && b.isMinus) return false; if(!isMinus && !b.isMinus){ if(V.size() < b.V.size()) return true; if(V.size() > b.V.size()) return false; for(int i=V.size(); i--;){ if(V.at(i) == b.V.at(i)) continue; return V.at(i) < b.V.at(i); } return false; } else{ if(V.size() > b.V.size()) return true; if(V.size() < b.V.size()) return false; for(int i=V.size(); i--;){ if(V.at(i) == b.V.at(i)) continue; return V.at(i) > b.V.at(i); } return false; } } bool operator<=(const bigint &b){ if(isMinus && !b.isMinus) return true; if(!isMinus && b.isMinus) return false; if(!isMinus && !b.isMinus){ if(V.size() < b.V.size()) return true; if(V.size() > b.V.size()) return false; for(int i=V.size(); i--;){ if(V.at(i) == b.V.at(i)) continue; return V.at(i) < b.V.at(i); } return true; } else{ if(V.size() > b.V.size()) return true; if(V.size() < b.V.size()) return false; for(int i=V.size(); i--;){ if(V.at(i) == b.V.at(i)) continue; return V.at(i) > b.V.at(i); } return true; } } bool operator>(const bigint &b){return !(*this<=b);} bool operator>=(const bigint &b){return !(*this>(istream &is,bigint &a){ string s; cin >> s; a = bigint(s); return is; } ostream &operator<<(ostream &os,bigint a){ if(a.isMinus) cout << "-"; if(a.V.size() == 0) cout << 0; for(int i=a.V.size(); i--;) cout << a.V.at(i); return os; } int main(){ ios_base::sync_with_stdio(false); cin.tie(nullptr); int T; cin >> T; int Need = 100000; vector allp; vector prime(Need+1,true); prime.at(0) = false; prime.at(1) = false; for(int i=2; i<=Need; i++){ if(!prime.at(i)) continue; allp.push_back(i); for(long long k=((long long)i)*i; k<=Need; k+=i) prime.at(k) = false; } while(T--){ bigint a,b,c; cin >> a >> b >> c; mint answer = 0; while(a%c == 0) answer += b.get(),a /= c; long long a1 = 0,a2 = 0,a3 = 0,p15 = 1'000'000'000'000'000LL; long long c1 = 0,c2 = 0,c3 = 0; auto divA = [&](int p) -> void { a /= p; while(a.V.size() <= 40) a.V.push_back(0); a1 = 0,a2 = 0,a3 = 0; for(int i=14; i>=0; i--){ a1 *= 10,a1 += a.V.at(i); } for(int i=29; i>=15; i--){ a2 *= 10,a2 += a.V.at(i); } for(int i=40; i>=30; i--){ a3 *= 10,a3 += a.V.at(i); } while(a.V.back() == 0) a.V.pop_back(); }; auto divC = [&](int p) -> void { c /= p; while(c.V.size() <= 40) c.V.push_back(0); c1 = 0,c2 = 0,c3 = 0; for(int i=14; i>=0; i--){ c1 *= 10,c1 += c.V.at(i); } for(int i=29; i>=15; i--){ c2 *= 10,c2 += c.V.at(i); } for(int i=40; i>=30; i--){ c3 *= 10,c3 += c.V.at(i); } while(c.V.back() == 0) c.V.pop_back(); }; divA(1),divC(1); pair worst = {1,1}; auto chmin = [&](pair check){if(worst.first*check.second > worst.second*check.first) worst = check;}; for(auto p : allp){ int one = 0,two = 0; while((a1%p+a2%p*(p15%p)+a3%p*(p15%p)*(p15%p))%p == 0) divA(p),one++; while((c1%p+c2%p*(p15%p)+c3%p*(p15%p)*(p15%p))%p == 0) divC(p),two++; if(two == 0) continue; chmin({one,two}); if(one == 0) break; } if(worst.first != 0 && c != 1){ bigint a2 = 1; vector plus1; for(int i=0; i<20; i++){ a2 *= a; if(a2%c == 0) a2 /= c,plus1.push_back(1); else plus1.push_back(0); } for(int i=2; i<=7; i++) for(int k=1; k=i)!=plus1.at(p)) ok = false; s %= i; } if(ok) chmin({k,i}); } } bigint b2 = b/worst.second*worst.first; int bint = (b%worst.second).get().v; answer += b2.get(); for(int i=1; ; i++){ if(i*worst.second <= bint*worst.first) answer++; else break; } cout << answer.v << "\n"; } }