(HDU)How many integers can you find（容斥原理 -好题）

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How many integers can you find

Time Limit: 12000/5000 MS (Java/Others)    Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 8740    Accepted Submission(s): 2594

Problem Description

  Now you get a number N, and a M-integers set, you should find out how many integers which are small than N, that they can divided exactly by any integers in the set. For example, N=12, and M-integer set is {2,3}, so there is another set {2,3,4,6,8,9,10}, all the integers of the set can be divided exactly by 2 or 3. As a result, you just output the number 7.

 

Input

  There are a lot of cases. For each case, the first line contains two integers N and M. The follow line contains the M integers, and all of them are different from each other. 0<N<2^31,0<M<=10, and the M integer are non-negative and won’t exceed 20.

 

Output

  For each case, output the number.

 

Sample Input

12 22 3

 

Sample Output

7

 

Author

wangye

 

Source

2008 “Insigma International Cup” Zhejiang Collegiate Programming Contest - Warm Up（4）

 

题意：找出从1至数字N里面有几个能整除M数组里面的数字。注：不包含N。

题解 ：1.M数组里面可能为0,和大于N的数字，在输入时应当直接排除。

     2.M数组里面有可能不全部互质，要用Gcd和lcm求最大公倍数来代替被除数。

             3.利用容斥原理进行“奇加偶减”。

代码：

#include<iostream>#include<algorithm>#include<cstdio>using namespace std;int n;int m;int a[15];int gcd(int a,int b){//gcd模板return a==0?b:gcd(b%a,a);}int lcm(int a,int b){//lcm模板return a/gcd(a,b)*b;}int main(){while(cin>>n>>m){n=n-1;int temp;for(int i=0;i<m;i++){cin>>temp;if(temp>0&&temp<=n)//判断输入是否符合大于0 ，小于na[i]=temp;else{m--;i--;}}int ans=0;for(int i=1;i<(1<<m);i++){//容斥原理，利用二进制进行列举情况int ant=0;int ride=1;for(int j=0;j<m;j++){if(i & (1<<j)){//哪一个数被选作被除数ant++;//记录被除因子的个数ride=lcm(ride,a[j]);//利用lcm求积，不能直接相乘}}if(ant & 1)//是否为奇数ans+=n/ride;elseans-=n/ride;}cout<<ans<<endl;}return 0;}

How many integers can you find

Time Limit: 12000/5000 MS (Java/Others)    Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 8740    Accepted Submission(s): 2594

Problem Description

  Now you get a number N, and a M-integers set, you should find out how many integers which are small than N, that they can divided exactly by any integers in the set. For example, N=12, and M-integer set is {2,3}, so there is another set {2,3,4,6,8,9,10}, all the integers of the set can be divided exactly by 2 or 3. As a result, you just output the number 7.

 

Input

  There are a lot of cases. For each case, the first line contains two integers N and M. The follow line contains the M integers, and all of them are different from each other. 0<N<2^31,0<M<=10, and the M integer are non-negative and won’t exceed 20.

 

Output

  For each case, output the number.

 

Sample Input

12 22 3

 

Sample Output

7

 

Author

wangye

 

Source

2008 “Insigma International Cup” Zhejiang Collegiate Programming Contest - Warm Up（4）