Mathematical knowledge: divisible number inclusion exclusion principle

subject ：AcWing 890. The number divisible
Given an integer n and m A different prime number p₁,p₂,…,p_m.

Please find out 1∼n Middle energy quilt p₁,p₂,…,p_m How many integers are divided by at least one number in .

Input format
The first line contains integers n and m.

The second line contains m A prime number .

Output format
Output an integer , Represents the number of integers that satisfy the condition .

Data range
1≤m≤16,
1≤n,p_i≤10⁹

sample input ：

10 2
2 3

sample output ：

7

Topic analysis ：
This question uses Principle of tolerance and exclusion
First of all, it can be used m Division of different prime numbers , Then maybe there will be 1、2、3… m Circumstances .
We use binary numbers to enumerate all cases
use S₁ Represents the number of sets that can only be divided by one number ,S₂ The number of sets that can be divided by two numbers …
According to the principle of inclusion and exclusion, all cases are set with odd number minus even number .

#include <iostream>

using namespace std;
typedef long long LL;
const int N = 20;

int p[N];

int main()
{
    
    int n,m;
    cin>>n>>m;
    for(int i=0;i<m;i++)cin>>p[i];
    
    int res=0;
    for(int i=1;i<1<<m;i++) // Enumerate all the situations 
    {
    
        int t=1,s=0; // t Represents the product of numbers that can be divisible ,s The number of numbers 
        for(int j=0;j<m;j++)
        {
    
            if(i>>j&1) //  If selected 
            {
    
                if((LL)t*p[j]>n) //  If the selected number is greater than n
                {
    
                    t=-1;
                    break;
                }
                t *= p[j];
                s++;
            }
        }
        if(t!=-1)
        {
    
            if(s%2)res+=n/t; // Odd plus 
            else res-=n/t; //  Even minus 
        }
    }
    cout<<res<<endl;
    return 0;
}