#pragma GCC optimize("O3") #pragma GCC optimize("unroll-loops") #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; //#define int long long typedef long long ll; typedef unsigned long long ul; typedef unsigned int ui; constexpr ll mod = 998244353; //constexpr ll mod = 1000000007; const ll INF = mod * mod; typedef pairP; #define rep(i,n) for(int i=0;i=0;i--) #define Rep(i,sta,n) for(int i=sta;i=1;i--) #define Rep1(i,sta,n) for(int i=sta;i<=n;i++) #define all(v) (v).begin(),(v).end() typedef pair LP; template void chmin(T& a, T b) { a = min(a, b); } template void chmax(T& a, T b) { a = max(a, b); } template void cinarray(vector& v) { rep(i, v.size())cin >> v[i]; } template void coutarray(vector& v) { rep(i, v.size()) { if (i > 0)cout << " "; cout << v[i]; } cout << "\n"; } ll mod_pow(ll x, ll n, ll m = mod) { if (n < 0) { ll res = mod_pow(x, -n, m); return mod_pow(res, m - 2, m); } if (abs(x) >= m)x %= m; if (x < 0)x += m; //if (x == 0)return 0; ll res = 1; while (n) { if (n & 1)res = res * x % m; x = x * x % m; n >>= 1; } return res; } struct modint { int n; modint() :n(0) { ; } modint(ll m) { if (m < 0 || mod <= m) { m %= mod; if (m < 0)m += mod; } n = m; } operator int() { return n; } }; bool operator==(modint a, modint b) { return a.n == b.n; } bool operator<(modint a, modint b) { return a.n < b.n; } modint operator+=(modint& a, modint b) { a.n += b.n; if (a.n >= mod)a.n -= mod; return a; } modint operator-=(modint& a, modint b) { a.n -= b.n; if (a.n < 0)a.n += mod; return a; } modint operator*=(modint& a, modint b) { a.n = ((ll)a.n * b.n) % mod; return a; } modint operator+(modint a, modint b) { return a += b; } modint operator-(modint a, modint b) { return a -= b; } modint operator*(modint a, modint b) { return a *= b; } modint operator^(modint a, ll n) { if (n == 0)return modint(1); modint res = (a * a) ^ (n / 2); if (n % 2)res = res * a; return res; } ll inv(ll a, ll p) { return (a == 1 ? 1 : (1 - p * inv(p % a, a)) / a + p); } modint operator/(modint a, modint b) { return a * modint(inv(b, mod)); } modint operator/=(modint& a, modint b) { a = a / b; return a; } const int max_n = 1 << 20; modint fact[max_n], factinv[max_n]; void init_f() { fact[0] = modint(1); for (int i = 0; i < max_n - 1; i++) { fact[i + 1] = fact[i] * modint(i + 1); } factinv[max_n - 1] = modint(1) / fact[max_n - 1]; for (int i = max_n - 2; i >= 0; i--) { factinv[i] = factinv[i + 1] * modint(i + 1); } } modint comb(int a, int b) { if (a < 0 || b < 0 || a < b)return 0; return fact[a] * factinv[b] * factinv[a - b]; } modint combP(int a, int b) { if (a < 0 || b < 0 || a < b)return 0; return fact[a] * factinv[a - b]; } ll gcd(ll a, ll b) { a = abs(a); b = abs(b); if (a < b)swap(a, b); while (b) { ll r = a % b; a = b; b = r; } return a; } typedef long double ld; typedef pair LDP; const ld eps = 1e-8; const ld pi = acosl(-1.0); template void addv(vector& v, int loc, T val) { if (loc >= v.size())v.resize(loc + 1, 0); v[loc] += val; } /*const int mn = 100005; bool isp[mn]; vector ps; void init() { fill(isp + 2, isp + mn, true); for (int i = 2; i < mn; i++) { if (!isp[i])continue; ps.push_back(i); for (int j = 2 * i; j < mn; j += i) { isp[j] = false; } } }*/ //[,val) template auto prev_itr(set& st, T val) { auto res = st.lower_bound(val); if (res == st.begin())return st.end(); res--; return res; } //[val,) template auto next_itr(set& st, T val) { auto res = st.lower_bound(val); return res; } using mP = pair; const string drul = "DRUL"; //DRUL int dx[4] = { 1,0,-1,0 }; int dy[4] = { 0,1,0,-1 }; //----------------------------------------- //x,yがax+by=gcd(a,b)の解になる ll extgcd(ll a, ll b, ll& x, ll& y) { ll d = a; if (b != 0) { d = extgcd(b, a % b, y, x); y -= (a / b) * x; } else { x = 1; y = 0; } return d; } //aのmod mでの逆元を求める ll mod_inverse(ll a, ll m) { ll x, y; extgcd(a, m, x, y); return (m + x % m) % m; } const int mn = 1 << 18; bool isp[mn]; int chp[mn]; void init() { fill(isp + 2, isp + mn, true); for (int i = 2; i < mn; i++) { if (!isp[i])continue; chp[i] = i; for (int j = 2 * i; j < mn; j += i) { isp[j] = false; chp[j] = i; } } } vector

resol(int x) { vector

res; while (x > 1) { int p = chp[x]; int cnt = 0; while (x % p == 0) { x /= p; cnt++; } res.push_back({ p,cnt }); } return res; } void solve() { int n; cin >> n; vector a(n), b(n); rep(i, n)cin >> a[i] >> b[i]; vector cur(mn); modint sa = 0, sb = 1; rep(i, n) { vector

ps = resol(b[i]); modint cr = sb; modint cl = 1; for (auto p : ps) { if (cur[p.first] > p.second) { cr *= mod_pow(p.first, -p.second); } else if (cur[p.first] <= p.second) { cr *= mod_pow(p.first, -cur[p.first]); cl *= mod_pow(p.first, p.second - cur[p.first]); } } sa = cl * sa + cr * (modint)a[i]; for (auto p : ps) { if (cur[p.first] < p.second) { sb *= mod_pow(p.first, p.second - cur[p.first]); cur[p.first] = p.second; } } } vector> loc(mn); rep(i, n) { auto ps = resol(b[i]); for (auto p : ps)loc[p.first].push_back(i); } rep(i, mn)if (cur[i] > 0) { ll pt = 1; rep(j, cur[i])pt *= i; ll x = 0; for (int id : loc[i]) { int cnt = 0; int cop = b[id]; while (cop % i == 0) { cop /= i; cnt++; } ll val = mod_inverse(cop, pt) * a[id]; val %= pt; rep(j, cur[i] - cnt) { val *= i; val %= pt; } x += val; x %= pt; } int z = 0; if (x == 0)z = cur[i]; else { while (x % i == 0) { x /= i; z++; } } sa /= mod_pow(i, z); sb /= mod_pow(i, z); } cout << sa << " " << sb << "\n"; } signed main() { ios::sync_with_stdio(false); cin.tie(0); //cout << fixed << setprecision(10); //init_f(); init(); //while(true) //expr(); //int t; cin >> t; rep(i, t) solve(); return 0; }