#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #define rep(i,n) for(int (i)=0;(i)<(int)(n);++(i)) #define rer(i,l,u) for(int (i)=(int)(l);(i)<=(int)(u);++(i)) #define reu(i,l,u) for(int (i)=(int)(l);(i)<(int)(u);++(i)) #if defined(_MSC_VER) || __cplusplus > 199711L #define aut(r,v) auto r = (v) #else #define aut(r,v) __typeof(v) r = (v) #endif #define each(it,o) for(aut(it, (o).begin()); it != (o).end(); ++ it) #define all(o) (o).begin(), (o).end() #define pb(x) push_back(x) #define mp(x,y) make_pair((x),(y)) #define mset(m,v) memset(m,v,sizeof(m)) #define INF 0x3f3f3f3f #define INFL 0x3f3f3f3f3f3f3f3fLL using namespace std; typedef vector vi; typedef pair pii; typedef vector > vpii; typedef long long ll; template inline void amin(T &x, U y) { if(y < x) x = y; } template inline void amax(T &x, U y) { if(x < y) x = y; } vector isprime; vector primes; void sieve(int n){ if((int)isprime.size() >= n+1) return; isprime.assign(n+1, true); isprime[0] = isprime[1] = false; int sqrtn = (int)(sqrt(n * 1.) + .5); for(int i = 2; i <= sqrtn; i ++) if(isprime[i]) { for(int j = i * i; j <= n; j += i) isprime[j] = false; } primes.clear(); for(int i = 2; i <= n; i ++) if(isprime[i]) primes.push_back(i); } typedef int FactorsInt; typedef vector > Factors; void primeFactors(FactorsInt x, Factors &out_v) { out_v.clear(); int sqrtx = (int)(sqrt(x*1.) + 10.5); sieve(sqrtx); for(vector::const_iterator p = primes.begin(); p != primes.end(); ++ p) { if(*p > sqrtx) break; if(x % *p == 0) { int t = 1; x /= *p; while(x % *p == 0) { t ++; x /= *p; } out_v.push_back(make_pair(*p, t)); } } if(x != 1) out_v.push_back(make_pair(x, 1)); } template bool expandLCE(const vector > &equations, vector > &res) { int n = equations.size(); Factors fs; res.clear(); map > a; for(int i = 0; i < n; ++ i) { int x = equations[i].first; primeFactors(equations[i].second, fs); for(int j = 0; j < (int)fs.size(); ++ j) { int p = fs[j].first, q = fs[j].second, pq = p; for(int k = 1; k < q; ++ k) pq *= p; pair &t = a[p]; if(t.first == 0) t.first = 1; Num u = min(t.first, pq); if((t.second - x) % u != 0) return false; if(t.first < pq) { t.first = pq; t.second = x % pq; } } } for(typename map >::iterator it = a.begin(); it != a.end(); ++ it) res.push_back(make_pair(it->second.first, it->second.second)); return true; } int inverse(signed a, const int MOD) { a %= MOD; if(a < 0) a += MOD; signed b = MOD, u = 1, v = 0; while(b) { signed t = a / b; a -= t * b; swap(a, b); u -= t * v; swap(u, v); } if(u < 0) u += MOD; return u; } //prefix[i] = CRT(pairs[0..i-1]) mod pairs[i].first を求める // //pairsの一番後ろに(Mod,-1)を追加しておくことによって // CRT(pairs) mod Mod を求めることができる //gcd(pairs[i].first, pairs[j].first) == 1 for i≠j を仮定する //さらに、pairs[i].first > 1 も仮定する! void prefixCRT(const vector > &pairs, vector &prefixes) { int n = pairs.size(); vector ts(n); for(int i = 0; i < n; ++ i) { int m = pairs[i].first, x = 1; for(int j = 0; j < i; ++ j) x = (long long)x * pairs[j].first % m; ts[i] = inverse(x, m); } prefixes.resize(n); for(int i = 0; i < n; ++ i) { int m = pairs[i].first; int n1 = 1, a = 0; for(int j = 0; j < i; ++ j) { int n2 = pairs[j].first, a2 = pairs[j].second; int h = (long long)(a2 - prefixes[j]) * ts[j] % n2; if(h < 0) h += n2; a = (a + (long long)n1 * h) % m; n1 = (long long)n1 * pairs[j].first % m; } prefixes[i] = a; } } int main() { sieve(31623); int N; scanf("%d", &N); vector equations; rep(i, N) { int X, Y; cin >> X >> Y; equations.push_back(mp(X, Y)); } vector > pairs; bool ok = expandLCE(equations, pairs); if(!ok) { puts("-1"); return 0; } const int Mod = 1000000007; pairs.push_back(make_pair(Mod, -1)); vector prefixes; prefixCRT(pairs, prefixes); int ans = prefixes.back(); bool zero = true; rep(i, pairs.size() - 1) zero &= pairs[i].second == 0; if(zero) { int prod = 1; rep(i, pairs.size() - 1) prod = (ll)prod * pairs[i].first % Mod; ans += prod; } printf("%d\n", ans); return 0; }