1306

1306 - Solutions to an Equation

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Time Limit: 2 second(s)

Memory Limit: 32 MB

You have to find the number of solutions of the following equation:

Ax + By + C = 0

Where A, B, C, x, y are integers and x₁ ≤ x ≤ x₂ and y₁ ≤ y ≤ y₂.

Input

Input starts with an integer T (≤ 10000), denoting the number of test cases.

Each case starts with a line containing seven integers A, B, C, x₁, x₂, y₁, y₂ (x₁ ≤ x₂, y₁ ≤ y₂). The value of each integer will lie in the range [-10⁸, 10⁸].

Output

For each case, print the case number and the total number of solutions.

Sample Input	Output for Sample Input
5 1 1 -5 -5 10 2 4 -10 -8 80 -100 100 -90 90 2 3 -4 1 7 0 8 -2 -3 6 -2 5 -10 5 1 8 -32 0 0 1 10	Case 1: 3 Case 2: 37 Case 3: 1 Case 4: 2 Case 5: 1

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PROBLEM SETTER: JANE ALAM JAN

思路：扩展欧几里得；

这题再次让我重新认识了扩欧；其实这题只要分情况讨论下，用扩欧求出特解，然后二分找通解中

t 的范围然后看t重合的部分就行，有几个特殊情况讨论下，还有A，B系数的符号讨论下。

#include<stdio.h>
#include<algorithm>
#include<iostream>
#include<string.h>
#include<queue>
#include<stack>
#include<set>
#include<math.h>
using namespace std;
typedef long long LL;
LL  gcd(LL n,LL m)
{
    if(m==0)
    {
        return n;
    }
    else if(n%m==0)
    {
        return m;
    }
    else return gcd(m,n%m);
}
pair<LL,LL>P(LL n,LL m)
{
    if(m==0)
    {
        pair<LL,LL>ask=make_pair(1,0);
        return ask;
    }
    else
    {
        pair<LL,LL>an=P(m,n%m);
        LL x=an.second;
        LL y=an.first;
        y-=(n/m)*x;
        an.first=x;
        an.second=y;
        return an;
    }
}
int main(void)
{
    LL i,j,k;
    scanf("%lld",&k);
    LL s;
    LL A,B,C,x1,x2,y1,y2;
    for(s=1; s<=k; s++)
    {
        LL sum=0;
        scanf("%lld %lld %lld %lld %lld %lld %lld",&A,&B,&C,&x1,&x2,&y1,&y2);
        C=-C;
        if(A==0&&B==0&&C!=0)
            sum=0;
        else if(A==0&&B==0&&C==0)
        {
            sum=(LL)(x2-x1+1)*(LL)(y2-y1+1);
        }
        else if(A==0)
        {
            if(C%B)
            {
                sum=0;
            }
            else
            {
                LL t=(C/B);
                if(t>=y1&&t<=y2)
                    sum=(x2-x1+1);
                else sum=0;
            }
        }
        else if(B==0)
        {
            if(C%A)
            {
                sum=0;
            }
            else
            {
                LL t=(C/A);
                if(t>=x1&&t<=x2)
                    sum=(y2-y1+1);
                else sum=0;
            }
        }
        else
        {
            if(A<0)
            {
                C=-C;
                A=-A;
                B=-B;
            }
            LL gc=gcd(abs(A),abs(B));
            if(C%gc)
            {
                sum=0;
            }
            else  if((LL)A*(LL)B>0)
            {
                A/=gc;
                B/=gc;
                C/=gc;
                pair<LL,LL>ask=P((A),(B));
                LL x=(LL)ask.first;
                LL y=(LL)ask.second;
                x*=C;
                y*=C;
                LL l=-1e9;
                LL r=1e9;
                LL id=1e9;
                while(l<=r)
                {
                    LL mid=(l+r)/2;
                    if(x+mid*B>=x1)
                    {
                        id=mid;
                        r=mid-1;
                    }
                    else l=mid+1;
                }
                l=-1e9;
                r=1e9;
                LL ic=1e9;
                while(l<=r)
                {
                    LL mid=(l+r)/2;
                    if(x+mid*B<=x2)
                    {
                        ic=mid;
                        l=mid+1;
                    }
                    else r=mid-1;
                }
                if(id>ic)
                {
                    sum=0;
                }
                else if(id==ic)
                {
                    LL xx=x+id*B;
                    if(xx>=x1&&xx<=x2)
                    {
                        LL yy=y-id*A;
                        if(yy>=y1&&yy<=y2)
                        {
                            sum=1;
                        }
                    }
                    else sum=0;
                }
                else
                {
                    l=-1e9;
                    r=1e9;
                    LL ip=1e9,iq=1e9;
                    while(l<=r)
                    {
                        LL mid=(l+r)/2;
                        if(y-mid*A>=y1)
                        {
                            ip=mid;
                            l=mid+1;
                        }
                        else r=mid-1;
                    }
                    l=-1e9;
                    r=1e9;
                    while(l<=r)
                    {
                        LL mid=(l+r)/2;
                        if(y-mid*A<=y2)
                        {
                            iq=mid;
                            r=mid-1;
                        }
                        else l=mid+1;
                    }
                    if(ip<iq)
                    {
                        sum=0;
                    }
                    else
                    {

                        if(ic<iq||id>ip)
                        {
                            sum=0;
                        }
                        else
                        {
                            if(id<=iq&&ic>=ip)
                            {
                                sum=ip-iq+1;
                            }
                            else if(iq<=id&&ip>=ic)
                            {
                                sum=ic-id+1;
                            }
                            else if(iq>=id&&iq<=ic)
                            {
                                sum=ic-iq+1;
                            }
                            else if(id>=iq&&id<=ip)
                            {
                                sum=ip-id+1;

                            }
                        }
                    }
                }
            }
            else
            {
                A/=gc;
                B/=gc;
                C/=gc;
                pair<LL,LL>ask=P(abs(A),abs(B));
                LL x=(LL)ask.first;
                LL y=(LL)ask.second;
                y=-y;
                x*=C;
                y*=C;
                LL l=-1e9;
                LL r=1e9;
                LL id=1e9;
                while(l<=r)
                {
                    LL mid=(l+r)/2;
                    if(x+mid*abs(B)>=x1)
                    {
                        id=mid;
                        r=mid-1;
                    }
                    else l=mid+1;
                }
                l=-1e9;
                r=1e9;
                LL ic=1e9;
                while(l<=r)
                {
                    LL mid=(l+r)/2;
                    if(x+mid*abs(B)<=x2)
                    {
                        ic=mid;
                        l=mid+1;
                    }
                    else r=mid-1;
                }
                if(id>ic)
                {
                    sum=0;
                }
                else if(id==ic)
                {
                    LL xx=x+id*abs(B);
                    if(xx>=x1&&xx<=x2)
                    {
                        LL yy=y+id*A;
                        if(yy>=y1&&yy<=y2)
                        {
                            sum=1;
                        }
                    }
                    else sum=0;
                }
                else
                {
                    l=-1e9;
                    r=1e9;
                    LL ip=1e9,iq=1e9;
                    while(l<=r)
                    {
                        LL mid=(l+r)/2;
                        if(y+mid*A>=y1)
                        {
                            iq=mid;
                            r=mid-1;
                        }
                        else l=mid+1;
                    }
                    l=-1e9;
                    r=1e9;
                    while(l<=r)
                    {
                        LL mid=(l+r)/2;
                        if(y+mid*A<=y2)
                        {
                            ip=mid;
                            l=mid+1;
                        }
                        else r=mid-1;
                    }
                    if(ip<iq)
                    {
                        sum=0;
                    }
                    else
                    {

                        if(ic<iq||id>ip)
                        {
                            sum=0;
                        }
                        else
                        {
                            if(id<=iq&&ic>=ip)
                            {
                                sum=ip-iq+1;
                            }
                            else if(iq<=id&&ip>=ic)
                            {
                                sum=ic-id+1;
                            }
                            else if(iq>=id&&iq<=ic)
                            {
                                sum=ic-iq+1;
                            }
                            else if(id>=iq&&id<=ip)
                            {
                                sum=ip-id+1;

                            }
                        }
                    }//printf("%lld %lld %lld %lld
",ip,iq,id,ic);
                }
            }
        }
        printf("Case %lld: %lld
",s,sum);

    }
    return 0;
}