1306 - Solutions to an Equation
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
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。
  1. #include<stdio.h>
  2. #include<algorithm>
  3. #include<iostream>
  4. #include<string.h>
  5. #include<queue>
  6. #include<stack>
  7. #include<set>
  8. #include<math.h>
  9. using namespace std;
  10. typedef long long LL;
  11. LL  gcd(LL n,LL m)
  12. {
  13.     if(m==0)
  14.     {
  15.         return n;
  16.     }
  17.     else if(n%m==0)
  18.     {
  19.         return m;
  20.     }
  21.     else return gcd(m,n%m);
  22. }
  23. pair<LL,LL>P(LL n,LL m)
  24. {
  25.     if(m==0)
  26.     {
  27.         pair<LL,LL>ask=make_pair(1,0);
  28.         return ask;
  29.     }
  30.     else
  31.     {
  32.         pair<LL,LL>an=P(m,n%m);
  33.         LL x=an.second;
  34.         LL y=an.first;
  35.         y-=(n/m)*x;
  36.         an.first=x;
  37.         an.second=y;
  38.         return an;
  39.     }
  40. }
  41. int main(void)
  42. {
  43.     LL i,j,k;
  44.     scanf("%lld",&k);
  45.     LL s;
  46.     LL A,B,C,x1,x2,y1,y2;
  47.     for(s=1; s<=k; s++)
  48.     {
  49.         LL sum=0;
  50.         scanf("%lld %lld %lld %lld %lld %lld %lld",&A,&B,&C,&x1,&x2,&y1,&y2);
  51.         C=-C;
  52.         if(A==0&&B==0&&C!=0)
  53.             sum=0;
  54.         else if(A==0&&B==0&&C==0)
  55.         {
  56.             sum=(LL)(x2-x1+1)*(LL)(y2-y1+1);
  57.         }
  58.         else if(A==0)
  59.         {
  60.             if(C%B)
  61.             {
  62.                 sum=0;
  63.             }
  64.             else
  65.             {
  66.                 LL t=(C/B);
  67.                 if(t>=y1&&t<=y2)
  68.                     sum=(x2-x1+1);
  69.                 else sum=0;
  70.             }
  71.         }
  72.         else if(B==0)
  73.         {
  74.             if(C%A)
  75.             {
  76.                 sum=0;
  77.             }
  78.             else
  79.             {
  80.                 LL t=(C/A);
  81.                 if(t>=x1&&t<=x2)
  82.                     sum=(y2-y1+1);
  83.                 else sum=0;
  84.             }
  85.         }
  86.         else
  87.         {
  88.             if(A<0)
  89.             {
  90.                 C=-C;
  91.                 A=-A;
  92.                 B=-B;
  93.             }
  94.             LL gc=gcd(abs(A),abs(B));
  95.             if(C%gc)
  96.             {
  97.                 sum=0;
  98.             }
  99.             else  if((LL)A*(LL)B>0)
  100.             {
  101.                 A/=gc;
  102.                 B/=gc;
  103.                 C/=gc;
  104.                 pair<LL,LL>ask=P((A),(B));
  105.                 LL x=(LL)ask.first;
  106.                 LL y=(LL)ask.second;
  107.                 x*=C;
  108.                 y*=C;
  109.                 LL l=-1e9;
  110.                 LL r=1e9;
  111.                 LL id=1e9;
  112.                 while(l<=r)
  113.                 {
  114.                     LL mid=(l+r)/2;
  115.                     if(x+mid*B>=x1)
  116.                     {
  117.                         id=mid;
  118.                         r=mid-1;
  119.                     }
  120.                     else l=mid+1;
  121.                 }
  122.                 l=-1e9;
  123.                 r=1e9;
  124.                 LL ic=1e9;
  125.                 while(l<=r)
  126.                 {
  127.                     LL mid=(l+r)/2;
  128.                     if(x+mid*B<=x2)
  129.                     {
  130.                         ic=mid;
  131.                         l=mid+1;
  132.                     }
  133.                     else r=mid-1;
  134.                 }
  135.                 if(id>ic)
  136.                 {
  137.                     sum=0;
  138.                 }
  139.                 else if(id==ic)
  140.                 {
  141.                     LL xx=x+id*B;
  142.                     if(xx>=x1&&xx<=x2)
  143.                     {
  144.                         LL yy=y-id*A;
  145.                         if(yy>=y1&&yy<=y2)
  146.                         {
  147.                             sum=1;
  148.                         }
  149.                     }
  150.                     else sum=0;
  151.                 }
  152.                 else
  153.                 {
  154.                     l=-1e9;
  155.                     r=1e9;
  156.                     LL ip=1e9,iq=1e9;
  157.                     while(l<=r)
  158.                     {
  159.                         LL mid=(l+r)/2;
  160.                         if(y-mid*A>=y1)
  161.                         {
  162.                             ip=mid;
  163.                             l=mid+1;
  164.                         }
  165.                         else r=mid-1;
  166.                     }
  167.                     l=-1e9;
  168.                     r=1e9;
  169.                     while(l<=r)
  170.                     {
  171.                         LL mid=(l+r)/2;
  172.                         if(y-mid*A<=y2)
  173.                         {
  174.                             iq=mid;
  175.                             r=mid-1;
  176.                         }
  177.                         else l=mid+1;
  178.                     }
  179.                     if(ip<iq)
  180.                     {
  181.                         sum=0;
  182.                     }
  183.                     else
  184.                     {
  185.  
  186.                         if(ic<iq||id>ip)
  187.                         {
  188.                             sum=0;
  189.                         }
  190.                         else
  191.                         {
  192.                             if(id<=iq&&ic>=ip)
  193.                             {
  194.                                 sum=ip-iq+1;
  195.                             }
  196.                             else if(iq<=id&&ip>=ic)
  197.                             {
  198.                                 sum=ic-id+1;
  199.                             }
  200.                             else if(iq>=id&&iq<=ic)
  201.                             {
  202.                                 sum=ic-iq+1;
  203.                             }
  204.                             else if(id>=iq&&id<=ip)
  205.                             {
  206.                                 sum=ip-id+1;
  207.  
  208.                             }
  209.                         }
  210.                     }
  211.                 }
  212.             }
  213.             else
  214.             {
  215.                 A/=gc;
  216.                 B/=gc;
  217.                 C/=gc;
  218.                 pair<LL,LL>ask=P(abs(A),abs(B));
  219.                 LL x=(LL)ask.first;
  220.                 LL y=(LL)ask.second;
  221.                 y=-y;
  222.                 x*=C;
  223.                 y*=C;
  224.                 LL l=-1e9;
  225.                 LL r=1e9;
  226.                 LL id=1e9;
  227.                 while(l<=r)
  228.                 {
  229.                     LL mid=(l+r)/2;
  230.                     if(x+mid*abs(B)>=x1)
  231.                     {
  232.                         id=mid;
  233.                         r=mid-1;
  234.                     }
  235.                     else l=mid+1;
  236.                 }
  237.                 l=-1e9;
  238.                 r=1e9;
  239.                 LL ic=1e9;
  240.                 while(l<=r)
  241.                 {
  242.                     LL mid=(l+r)/2;
  243.                     if(x+mid*abs(B)<=x2)
  244.                     {
  245.                         ic=mid;
  246.                         l=mid+1;
  247.                     }
  248.                     else r=mid-1;
  249.                 }
  250.                 if(id>ic)
  251.                 {
  252.                     sum=0;
  253.                 }
  254.                 else if(id==ic)
  255.                 {
  256.                     LL xx=x+id*abs(B);
  257.                     if(xx>=x1&&xx<=x2)
  258.                     {
  259.                         LL yy=y+id*A;
  260.                         if(yy>=y1&&yy<=y2)
  261.                         {
  262.                             sum=1;
  263.                         }
  264.                     }
  265.                     else sum=0;
  266.                 }
  267.                 else
  268.                 {
  269.                     l=-1e9;
  270.                     r=1e9;
  271.                     LL ip=1e9,iq=1e9;
  272.                     while(l<=r)
  273.                     {
  274.                         LL mid=(l+r)/2;
  275.                         if(y+mid*A>=y1)
  276.                         {
  277.                             iq=mid;
  278.                             r=mid-1;
  279.                         }
  280.                         else l=mid+1;
  281.                     }
  282.                     l=-1e9;
  283.                     r=1e9;
  284.                     while(l<=r)
  285.                     {
  286.                         LL mid=(l+r)/2;
  287.                         if(y+mid*A<=y2)
  288.                         {
  289.                             ip=mid;
  290.                             l=mid+1;
  291.                         }
  292.                         else r=mid-1;
  293.                     }
  294.                     if(ip<iq)
  295.                     {
  296.                         sum=0;
  297.                     }
  298.                     else
  299.                     {
  300.  
  301.                         if(ic<iq||id>ip)
  302.                         {
  303.                             sum=0;
  304.                         }
  305.                         else
  306.                         {
  307.                             if(id<=iq&&ic>=ip)
  308.                             {
  309.                                 sum=ip-iq+1;
  310.                             }
  311.                             else if(iq<=id&&ip>=ic)
  312.                             {
  313.                                 sum=ic-id+1;
  314.                             }
  315.                             else if(iq>=id&&iq<=ic)
  316.                             {
  317.                                 sum=ic-iq+1;
  318.                             }
  319.                             else if(id>=iq&&id<=ip)
  320.                             {
  321.                                 sum=ip-id+1;
  322.  
  323.                             }
  324.                         }
  325.                     }//printf("%lld %lld %lld %lld\n",ip,iq,id,ic);
  326.                 }
  327.             }
  328.         }
  329.         printf("Case %lld: %lld\n",s,sum);
  330.  
  331.     }
  332.     return 0;
  333. }

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