[Knowledge] Chinese Remainder Theorem (CRT)

[Applications] solve simple linear congruent equation

 

 

[Board (between coprime m)]

/ * 
Chinese remainder theorem board (base) 
provided with a remainder of c [i], modulo m [i], let M m [i] sum and 
then the final answer is c [i] (M / m [i]) and modM inv ((m / m [ i]), m [i]) of the 
solution is unique, and because each of m mod are different, so that just cover all situations. 
* / 
LL C [MAXN], m [MAXN], M; // I from the beginning. 1 
LL exgcd (LL A, LL B, LL & X, LL & Y) 
{ 
    IF (! B) 
    { 
        X = . 1 , Y = 0 ;
         return A; 
    } 
    LL D = exgcd (B, A% B, Y, X); 
    Y - = A / B * X;
     return D; 
} 
LL getInv (LL A, LL MOD) 
{ 
    LL X, Y, D; 
    D =exgcd(a,mod,x,y);
    return d==1?(x%mod+mod)%mod:-1;
}
ll CRT(int n)
{
    M=1;
    for(int i=1;i<=n;i++)M*=m[i];
    ll ret=0;
    for(int i=1;i<=n;i++)
        ret=(ret+c[i] * (M/m[i]) %M * getInv(M/m[i],m[i]) )%M;
    return ret;
}

 

 

[Board (extended)]

/ * 
Chinese remainder theorem board (extension) 
number mo not prime to each of the time 
* / 
LL m [MAXN], C [MAXN]; // beginning. 1 
LL GCD (A LL, LL B) 
{ 
    return B! ? A: GCD (B, A% B); 
} 
LL exgcd (LL A, LL B, LL & X, LL & Y) 
{ 
    IF (! B) 
    { 
        X = . 1 , Y = 0 ;
         return A; 
    } 
    LL D exgcd = (B, A% B, Y, X); 
    Y - = A / B * X;
     return D; 
} 
LL getInv (A LL, LL MOD) 
{ 
    LL X, Y; 
    LL D=exgcd(a,mod,x,y);
    return d==1?(x+mod)%mod:-1;
}
ll exCRT(int n)
{
    ll m1,m2,c1,c2,d;
    for(int i=2;i<=n;i++)
    {
        m1=m[i-1],m2=m[i],c1=c[i-1],c2=c[i];
        d=gcd(m1,m2);
        if((c2-c1)%d)return -1;//此时无法合并
        m[i] = m[i-1] * m[i] / d;
        c[i] = (c2-c1)/d * getInv(m1/d,m2/d) % ( m2 / d ) * m1 + c1;
        c[i] = ( c[i] % m[i] + m[i] ) % m[i];
    }
    return c[n];
}

 

 

【prove】

I did not learn it, detailed look link

https://www.cnblogs.com/MashiroSky/p/5918158.html

 

【exercise】

hdu 1573

 

 

Reference [link]

https://blog.csdn.net/xiaoming_p/article/details/79672049