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HDU1695-GCD（数论-欧拉函数-容斥）

2024-07-15 03:29:01   223人阅读

GCD

Time Limit: 6000/3000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 5454    Accepted Submission(s): 1957


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).
 

题意： 求(1,a) 和(1,b) 两个区间 公约数为k的对数的个数

思路：将a,b分别处以k,就可以转化为(1,a/k)和(1,b/k)两个区间两两互质的个数，可以先用欧拉函数求出(1,a)两两互质的个数,(a+1，b) 可以分解质因数，因为质因数的个数最多为7可以用容斥原理计算。

#include <iostream>
#include <cstdio>
#include <cstring>
#include <vector>
#include <string>
#include <algorithm>
#include <queue>
using namespace std;

const int maxn = 10000+10;
const int maxxn = 100000+10;
typedef long long ll;
int a,b,gcd;
ll ans;
bool isPrime[maxn];
ll minDiv[maxxn],phi[maxxn],sum[maxxn];
vector<int> prime,cnt[maxxn],digit[maxxn];

void getPrime(){
    prime.clear();
    memset(isPrime,1,sizeof isPrime);
    for(int i = 2;i < maxn; i++){
        if(isPrime[i]){
            prime.push_back(i);
            for(int j = i*i; j < maxn; j+=i){
                isPrime[j] = 0;
            }
        }
    }
}

void getPhi(){
    for(ll i = 1; i < maxxn; i++){
        minDiv[i] = i;
    }
    for(ll i = 2; i*i < maxxn; i++){
        if(minDiv[i]==i){
            for(int j = i*i; j < maxxn; j += i){
                minDiv[j] = i;
            }
        }
    }
    phi[1] = 1;
    sum[1] = 1;
    for(ll i = 2; i < maxxn; i++){
        phi[i] = phi[i/minDiv[i]];
        if((i/minDiv[i])%minDiv[i]==0){
            phi[i] *= minDiv[i];
        }else{
            phi[i] *= minDiv[i]-1;
        }
        sum[i] = phi[i]+sum[i-1];
    }
}

void getDigit(){
    for(ll i = 1; i < maxxn; i++){
        int x = i;
        for(int j = 0; j < prime.size()&&x >= prime[j]; j++){
            if(x%prime[j]==0){
                digit[i].push_back(prime[j]);
                int t = 0;
                while(x%prime[j]==0){
                    t++;
                    x /= prime[j];
                }
                cnt[i].push_back(t);
            }
        }
        if(x!=1){
            digit[i].push_back(x);
            cnt[i].push_back(1);
        }
    }
}

int main(){
    getPrime();
    getPhi();
    getDigit();
    int ncase,T=1;
    cin >> ncase;
    while(ncase--){
        int t1,t2;
        scanf("%d%d%d%d%d",&t1,&a,&t2,&b,&gcd);
        if(gcd==0){
            printf("Case %d: 0\n",T++,ans);
            continue;
        }else{
            if(a > b) swap(a,b);
            a /= gcd,b /= gcd;
            ans = sum[a];
            for(ll i = a+1; i <= b; i++){
                int d = digit[i].size();
                int t = 0;
                vector<int> di;
                for(int k = 1; k < (1<<d); k++){
                    di.clear();
                    for(int f = 0; f < d; f++){
                        if(k&(1<<f)){
                            di.push_back(digit[i][f]);
                        }
                    }
                    int ji = 1;
                    for(int f = 0; f < di.size(); f++){
                        ji *= di[f];
                    }
                    if(di.size()%2==0){
                        t -= a/ji;
                    }else{
                        t += a/ji;
                    }
                }
                ans += a-t;
            }
            printf("Case %d: ",T++);
            cout<<ans<<endl;
        }

    }
    return 0;
}



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