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kattis_generalchineseremainder.cpp
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kattis_generalchineseremainder.cpp
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/**Kattis - generalchineseremainder
* Chinese remainder theorem with not necessarily co-prime moduli. We verify validity
* before we start trying to solve.
*
* Time: O(quite hard to analyse), Space: O(1)
*/
#pragma GCC optimize("Ofast")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,avx2,fma")
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef vector<ll> vll;
#define fast_cin() ios_base::sync_with_stdio(false); cin.tie(NULL); cout.tie(NULL);
ll fexp(ll b, ll p){
if (p == 0)return 1;
ll ans = fexp(b, p >> 1);
ans = ans * ans;
if (p & 1) ans = ans * b;
return ans;
}
ll _sieve_size;
bitset<10000010> bs;
vll primes;
void sieve(ll upperbound = (ll) 1e7){
_sieve_size = upperbound + 1;
bs.set();
bs[0] = bs[1] = 0;
for(ll i = 2; i <= _sieve_size; i++){
if(bs[i]){
for(ll j = i * i; j <= _sieve_size; j += i){
bs[j] = 0;
}
primes.push_back(i);
}
}
}
vector<tuple<ll,ll>> prime_factorise(ll n){
vector<tuple<ll, ll>> prime_factors;
int exp;
for(ll i = 0; i < (ll) primes.size() && primes[i] * primes[i] <= n; i++){
if(n % primes[i] == 0){
exp = 0;
while(n % primes[i] == 0){
n /= primes[i];
exp++;
}
prime_factors.emplace_back(primes[i], exp);
}
}
if(n != 1){
prime_factors.emplace_back(n, 1);
}
return prime_factors;
}
ll mod(ll a, ll n) { return (a % n + n) % n; }
ll extEuclid(ll a, ll b, ll &x, ll &y) { // pass x and y by ref
ll xx = y = 0;
ll yy = x = 1;
while (b) { // repeats until b == 0
ll q = a / b;
tie(a, b) = tuple(b, a % b);
tie(x, xx) = tuple(xx, x - q * xx);
tie(y, yy) = tuple(yy, y - q * yy);
}
return a; // returns gcd(a, b)
}
ll modInverse(ll a, ll n) { // returns modular inverse of a mod n
ll x, y;
extEuclid(a, n, x, y);
return mod(x, n);
}
ll crt(vll &r, vll &m) {
// m_t = m_0*m_1*...*m_{n-1}
ll mt = accumulate(m.begin(), m.end(), 1LL, multiplies<>()); // the LL is important!
ll x = 0;
for (int i = 0; i < (int)m.size(); ++i) {
ll a = mod((ll)r[i] * modInverse(mt / m[i], m[i]), m[i]);
x = mod(x + (ll)a * (mt / m[i]), mt);
}
return x;
}
bool verify_crt(vll &r, vll &m) {
// Ensure r_i = r_j (mod gcd(m_i, m_j)) for all i, j
for (int i = 0; i < (int)m.size(); ++i) {
for (int j = i+1; j < (int)m.size(); ++j) {
if (mod(r[i], gcd(m[i], m[j])) != mod(r[j], gcd(m[i], m[j]))) {
return false;
}
}
}
return true;
}
ll general_crt(vll r, vll m){
if (!verify_crt(r, m)) {
return -1;
}
unordered_map<ll, ll> prime_highest_pow;
unordered_map<ll, ll> S;
for (int i=0; i<(int)m.size(); ++i) {
vector<tuple<ll, ll>> factors = prime_factorise(m[i]);
for (int j=0; j<(int)factors.size(); ++j) {
auto &[p, exp] = factors[j];
if (prime_highest_pow[p] < exp){
prime_highest_pow[p] = exp;
S[p] = r[i] % (fexp(p, exp));
}
}
}
vll nr, nm;
for (auto &[p, exp] : prime_highest_pow) {
// cout << "p = " << p << " exp = " << exp << endl;
nr.push_back(S[p]);
nm.push_back(fexp(p, exp));
}
return crt(nr, nm);
}
int main(){
sieve();
int num_tc;
ll a, ni, b, mi;
cin >> num_tc;
while (num_tc--){
cin >> a >> ni >> b >> mi;
vll r = {a, b};
vll m = {ni, mi};
ll ans = general_crt(r, m);
if (ans == -1) {
cout << "no solution\n";
}
else{
cout << ans << " " << lcm(ni, mi) << endl;
}
}
return 0;
}