#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)); } 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; } int powpq(int p, int q) { int r = 1; rep(i, q) r *= p; return r; } int main() { sieve(31623); int N; scanf("%d", &N); Factors fs; map mods; bool ok = true; rep(i, N) { int X, Y; cin >> X >> Y; primeFactors(Y, fs); each(j, fs) { int p = j->first, q = j->second; int pq = powpq(p, q); pii &t = mods[p]; if(t.first == 0) t.first = 1; int u = min(pq, t.first); ok &= t.second % u == X % u; if(t.first < pq) { t.first = pq; t.second = X % pq; } } } if(!ok) { puts("-1"); return 0; } vector m, x; each(i, mods) { m.push_back(i->second.first); x.push_back(i->second.second); } m.push_back(1000000007); int P = m.size(); // rep(i, P-1) cerr << i << ": " << m[i] << ", " << x[i] << endl; vector prod(P, vi(P+1)); vector table(P, vi(P+1)); //table[i][j+1] //= CRT(x[0..j]) mod m[i] // let b = prod(m[0..j-1]) mod a2 // t = inverse(b) mod a2 // a1 = CRT(m[0..j-1]) mod a2 // a1' = CRT(m[0..j-1]) mod m[i] // h = (x[j] - a1) * t mod a2 // n1 = prod(m[0..j-1]) mod m[i] // in (a1' + n1 * h) mod m[i] // rep(i, P) { prod[i][0] = 1; rep(j, P) prod[i][j+1] = (ll)prod[i][j] * m[j] % m[i]; } rep(i, P) { table[i][0] = 0; int b = 1; rep(j, i) { int b = prod[j][j]; int t = inverse(b, m[j]); int a1 = table[j][j]; int a1m = table[i][j]; int h = (ll)(x[j] - a1) * t % m[j]; if(h < 0) h += m[j]; int n1 = prod[i][j]; table[i][j+1] = (a1m + (ll)n1 * h) % m[i]; // cerr << i << ", " << j << ": " << table[i][j+1] << endl; } } int ans = table[P-1][P-1]; bool zero = true; each(i, mods) zero &= i->second.second == 0; if(zero) ans += prod[P-1][P-1]; printf("%d\n", ans); return 0; }