HDU 1695 GCD （莫比烏斯反演）

GCD

Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 4291    Accepted Submission(s): 1502

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

 

 

 

前幾天用容斥原理寫過這題：

http://www.cnblogs.com/kuangbin/p/3269182.html

 

速度比較慢。

 

 

用莫比烏斯反演快很多。

 

莫比烏斯反演資料：

http://wenku.baidu.com/view/542961fdba0d4a7302763ad5.html

http://baike.baidu.com/link?url=1qQ-hkgOwDJAH4xyRcEQVoOTmHbiRCyZZ-hEJxRBQO8G0OurXNr6Rh6pYj9fhySI0MY2RKpcaSPV9X75mQv0hK

 

 

 

這題求[1,n],[1,m]gcd為k的對數。而且沒有順序。

轉化之后就是[1,n/k],[1,m/k]之間互質的數的個數。

 

用莫比烏斯反演就很容易求了。

 

 

為了去除重復的，去掉一部分就好了；

 

 

 

 

 

 

這題求的時候還可以分段進行優化的。

 

具體看我的下一篇博客吧！

 

 

 

/* ***********************************************
Author        :kuangbin
Created Time  :2013/8/21 19:32:35
File Name     :F:\2013ACM練習\專題學習\數學\莫比烏斯反演\HDU1695GCD.cpp
************************************************ */

#include <stdio.h>
#include <string.h>
#include <iostream>
#include <algorithm>
#include <vector>
#include <queue>
#include <set>
#include <map>
#include <string>
#include <math.h>
#include <stdlib.h>
#include <time.h>
using namespace std;
const int MAXN = 1000000;
bool check[MAXN+10];
int prime[MAXN+10];
int mu[MAXN+10];
void Moblus()
{
    memset(check,false,sizeof(check));
    mu[1] = 1;
    int tot = 0;
    for(int i = 2; i <= MAXN; i++)
    {
        if( !check[i] )
        {
            prime[tot++] = i;
            mu[i] = -1;
        }
        for(int j = 0; j < tot; j++)
        {
            if(i * prime[j] > MAXN) break;
            check[i * prime[j]] = true;
            if( i % prime[j] == 0)
            {
                mu[i * prime[j]] = 0;
                break;
            }
            else
            {
                mu[i * prime[j]] = -mu[i];
            }
        }
    }
}
int main()
{
    //freopen("in.txt","r",stdin);
    //freopen("out.txt","w",stdout);
    int T;
    int a,b,c,d,k;
    Moblus();
    scanf("%d",&T);
    int iCase = 0;
    while(T--)
    {
        iCase++;
        scanf("%d%d%d%d%d",&a,&b,&c,&d,&k);
        if(k == 0)
        {
            printf("Case %d: 0\n",iCase);
            continue;
        }
        b /= k;
        d /= k;
        if(b > d)swap(b,d);
        long long ans1 = 0;
        for(int i = 1; i <= b;i++)
            ans1 += (long long)mu[i]*(b/i)*(d/i);
        long long ans2 = 0;
        for(int i = 1;i <= b;i++)
            ans2 += (long long)mu[i]*(b/i)*(b/i);
        ans1 -= ans2/2;
        printf("Case %d: %I64d\n",iCase,ans1);
    }
    return 0;
}