GCD

Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 4272    Accepted Submission(s): 1492

Problem Description Given 5 integers: a, b, c, d, k, you’re to find x in a…b, y in c…d that GCD(x, y) = k. GCD(x, y) means the greatest common divisor of x and y. Since the number of choices may be very large, you’re only required to output the total number of different number pairs.

Please notice that, (x=5, y=7) and (x=7, y=5) are considered to be the same.

Yoiu can assume that a = c = 1 in all test cases.  

Input The input consists of several test cases. The first line of the input is the number of the cases. There are no more than 3,000 cases.

Each case contains five integers: a, b, c, d, k, 0 < a <= b <= 100,000, 0 < c <= d <= 100,000, 0 <= k <= 100,000, as described above.  

Output For each test case, print the number of choices. Use the format in the example.  

Sample Input 2 1 3 1 5 1 1 11014 1 14409 9  

Sample Output Case 1: 9 Case 2: 736427
Hint For the first sample input, all the 9 pairs of numbers are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 5), (3, 4), (3, 5).  

Source
2008 “Sunline Cup” National Invitational Contest  

Recommend wangye  

题意: 在1~a, 1~b中挑出(x,y)满足gcd(x,y) = k , 求(x,y) 的对数 , a,b<=10^5

思路: gcd(x, y) == k 说明x,y都能被k整除, 但是能被k整除的未必gcd=k  , 必须还要满足

互质关系. 问题就转化为了求1~a/k 和 1~b/k间互质对数的问题

可以把a设置为小的那个数, 那么以y>x来保持唯一性(题目要求, 比如[1,3] = [3,1] )

接下来份两种情况:

1. y <= a , 那么对数就是 1~a的欧拉函数的累计和(容易想到)

2. y >= a , 这个时候欧拉函数不能用了,怎么做?　　可以用容斥原理,把y与1~a互质对数问题转换为

  1 /* ***********************************************
  2 Author        :kuangbin
  3 Created Time  :2013/8/19 22:08:43
  4 File Name     :F:\2013ACM练习\专题学习\数学\HDU\HDU1695GCD.cpp
  5 ************************************************ */
  6 
  7 #include <stdio.h>
  8 #include <string.h>
  9 #include <iostream>
 10 #include <algorithm>
 11 #include <vector>
 12 #include <queue>
 13 #include <set>
 14 #include <map>
 15 #include <string>
 16 #include <math.h>
 17 #include <stdlib.h>
 18 #include <time.h>
 19 using namespace std;
 20 
 21 const int MAXN = 10000;
 22 int prime[MAXN+1];
 23 void getPrime()
 24 {
 25     memset(prime,0,sizeof(prime));
 26     for(int i = 2;i <= MAXN;i++)
 27     {
 28         if(!prime[i])prime[++prime[0]] = i;
 29         for(int j = 1;j <= prime[0] && prime[j] <= MAXN/i;j++)
 30         {
 31             prime[prime[j]*i] = 1;
 32             if(i%prime[j] == 0)break;
 33         }
 34     }
 35 }
 36 long long factor[100][2];
 37 int fatCnt;
 38 int getFactors(long long x)
 39 {
 40     fatCnt = 0;
 41     long long tmp = x;
 42     for(int i = 1; prime[i] <= tmp/prime[i];i++)
 43     {
 44         factor[fatCnt][1] = 0;
 45         if(tmp%prime[i] == 0)
 46         {
 47             factor[fatCnt][0] = prime[i];
 48             while(tmp%prime[i] == 0)
 49             {
 50                 factor[fatCnt][1]++;
 51                 tmp /= prime[i];
 52             }
 53             fatCnt++;
 54         }
 55     }
 56     if(tmp != 1)
 57     {
 58         factor[fatCnt][0] = tmp;
 59         factor[fatCnt++][1] = 1;
 60     }
 61     return fatCnt;
 62 }
 63 int euler[100010];
 64 void getEuler()
 65 {
 66     memset(euler,0,sizeof(euler));
 67     euler[1] = 1;
 68     for(int i = 2;i <= 100000;i++)
 69         if(!euler[i])
 70             for(int j = i; j <= 100000;j += i)
 71             {
 72                 if(!euler[j])
 73                     euler[j] = j;
 74                 euler[j] = euler[j]/i*(i-1);
 75             }
 76 }
 77 int calc(int n,int m)//n < m,求1-n内和m互质的数的个数
 78 {
 79     getFactors(m);
 80     int ans = 0;
 81     for(int i = 1;i < (1<<fatCnt);i++)
 82     {
 83         int cnt = 0;
 84         int tmp = 1;
 85         for(int j = 0;j < fatCnt;j++)
 86             if(i&(1<<j))
 87             {
 88                 cnt++;
 89                 tmp *= factor[j][0];
 90             }
 91         if(cnt&1)ans += n/tmp;
 92         else ans -= n/tmp;
 93     }
 94     return n - ans;
 95 }
 96 int main()
 97 {
 98     //freopen("in.txt","r",stdin);
 99     //freopen("out.txt","w",stdout);
100     getPrime();
101     int a,b,c,d;
102     int T;
103     int k;
104     scanf("%d",&T);
105     int iCase = 0;
106     getEuler();
107     while(T--)
108     {
109         iCase++;
110         scanf("%d%d%d%d%d",&a,&b,&c,&d,&k);
111         if(k == 0 || k > b || k > d)
112         {
113             printf("Case %d: 0\n",iCase);
114             continue;
115         }
116         if(b > d)swap(b,d);
117         b /= k;
118         d /= k;
119         long long ans = 0;
120         for(int i = 1;i <= b;i++)
121             ans += euler[i];
122         for(int i = b+1;i <= d;i++)
123             ans += calc(b,i);
124         printf("Case %d: %I64d\n",iCase,ans);
125     }
126     
127     return 0;
128 }