EXCRT(中国单身狗定理)

Posted 2021-01-31 llcsblog

(excrt) ( ext{是用来解决当模数不互质的情况下的普通}crt ext{情况的})

具体见P4777

事实上和(crt)没有什么关系

主要思想是:我们不断的合并两个同余方程，最后合并到只剩一个。

对于方程组

(egin{cases} x equiv b_1 ({ m mod} a_1) \\ xequiv b_2 ({ m mod} a_2) \\ end{cases})

我们考虑合并 这两个同余方程组

我们化为

(egin{cases} x =a_1k_1+b_1\\ x=a_2k_2+b_2 \\ end{cases})

( herefore a_1k_1+b_1=a_2k_2+b_2)

( herefore a_1k_1-a_2k_2=b_2-b_1)

这个可以用(exgcd)求出一组当(a_1k_1-a_2k_2=(a1,a2))时的特解

我们将(x*gcd(a1,a2)/(b_2-b_1))就可以得到原方程的一组解

那么我们可以回代解出(a_1,b_1,a_2,b_2)

然后

我们将(b_2-b_1)作为新的(x),([a_1,a_2])作为新的模数,(a_1k_1-a_2k_2)作为新的余数

我们就成功的合并了两个方程

然后不停的合并下去，最后的(x)就是答案了

代码

/*
@Date    : 2018-10-07 11:01:36
@Author  : Adscn ([email protected])
@Link    : https://www.cnblogs.com/LLCSBlog
*/
#ifdef FASTER
#pragma GCC diagnostic error "-std=c++11"
#pragma GCC optimize(2)
#pragma GCC optimize(3)
#pragma GCC optimize("Ofast")
#pragma GCC optimize("inline")
#pragma GCC optimize("-fgcse")
#pragma GCC optimize("-fgcse-lm")
#pragma GCC optimize("-fipa-sra")
#pragma GCC optimize("-ftree-pre")
#pragma GCC optimize("-ftree-vrp")
#pragma GCC optimize("-fpeephole2")
#pragma GCC optimize("-ffast-math")
#pragma GCC optimize("-fsched-spec")
#pragma GCC optimize("unroll-loops")
#pragma GCC optimize("-falign-jumps")
#pragma GCC optimize("-falign-loops")
#pragma GCC optimize("-falign-labels")
#pragma GCC optimize("-fdevirtualize")
#pragma GCC optimize("-fcaller-saves")
#pragma GCC optimize("-fcrossjumping")
#pragma GCC optimize("-fthread-jumps")
#pragma GCC optimize("-funroll-loops")
#pragma GCC optimize("-fwhole-program")
#pragma GCC optimize("-freorder-blocks")
#pragma GCC optimize("-fschedule-insns")
#pragma GCC optimize("inline-functions")
#pragma GCC optimize("-ftree-tail-merge")
#pragma GCC optimize("-fschedule-insns2")
#pragma GCC optimize("-fstrict-aliasing")
#pragma GCC optimize("-fstrict-overflow")
#pragma GCC optimize("-falign-functions")
#pragma GCC optimize("-fcse-skip-blocks")
#pragma GCC optimize("-fcse-follow-jumps")
#pragma GCC optimize("-fsched-interblock")
#pragma GCC optimize("-fpartial-inlining")
#pragma GCC optimize("no-stack-protector")
#pragma GCC optimize("-freorder-functions")
#pragma GCC optimize("-findirect-inlining")
#pragma GCC optimize("-fhoist-adjacent-loads")
#pragma GCC optimize("-frerun-cse-after-loop")
#pragma GCC optimize("inline-small-functions")
#pragma GCC optimize("-finline-small-functions")
#pragma GCC optimize("-ftree-switch-conversion")
#pragma GCC optimize("-foptimize-sibling-calls")
#pragma GCC optimize("-fexpensive-optimizations")
#pragma GCC optimize("-funsafe-loop-optimizations")
#pragma GCC optimize("inline-functions-called-once")
#pragma GCC optimize("-fdelete-null-pointer-checks")
#endif
#include<bits/stdc++.h>
using namespace std;
#define IL inline
#define RG register
#define gi getint()
#define gc getchar()
#define File(a) freopen(a".in","r",stdin);freopen(a".out","w",stdout)
typedef long long ll;
IL ll getint()
{
    RG ll xi=0;
    RG char ch=gc;
    bool f=0;
    while(ch<'0'|ch>'9')ch=='-'?f=1:f,ch=gc;
    while(ch>='0'&ch<='9')xi=(xi<<1)+(xi<<3)+ch-48,ch=gc;
    return f?-xi:xi;
}
inline ll exgcd(ll a,ll b,ll &x,ll &y){
    if(!b){
        x=1,y=0;
        return a;
    }
    ll gcd=exgcd(b,a%b,y,x);
    y-=(a/b)*x;
    return gcd;
}
struct node{
    ll mod,a;
}last,now;
istream& operator >> (istream&is,node &p)
{
    cin>>p.mod>>p.a;
    return is;
}
node join(node p,node q)
{
    ll x,y;
    ll gcd=exgcd(p.mod,q.mod,x,y);
    x=(x*(__int128)(q.a-p.a)/gcd)%(q.mod/gcd);
    if((q.a-p.a)%gcd)exit(1);
    p.a=x*p.mod+p.a;
    p.mod=p.mod/gcd*q.mod;
    p.a=(p.a%p.mod+p.mod)%p.mod;
    return p;
}
int main(void)
{
    #ifdef ONLINE_JUDGE
//    File("");
    #endif
    ll n;
    cin>>n>>last;
    for(int i=2;i<=n;i++)
    {
        cin>>now;
        last=join(last,now);
    }
    cout<<last.a;
    return 0;
}