Count pairs (i,j) such that (i+j) is divisible by A and B both

Given n, m, A and B. The task is to count the number of pairs of integers (x, y) such that 1 \leq x \leq n and 1 \leq y \leq m and (x+y) mod A and (x+y) mod B both equals to 0.

Examples:

Input: n = 60, m = 90, A = 5, B = 10
Output: 540

Input: n = 225, m = 452, A = 10, B = 15
Output: 3389

Approach: If (x+y) is divisible by both A and B then basically LCM of A and B is the smallest divisor of (x+y). So we calculate all numbers that is less than or equal to m and divisible by LCM of them and when iterating with the loop then we check if the present number is divisible by LCM of A and B.

Below is the implementation of the above approach:

C++

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// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the LCM
int find_LCM(int x, int y)
{
    return (x * y) / __gcd(x, y);
}
  
// Function to count the pairs
int CountPairs(int n, int m, int A, int B)
{
    int cnt = 0;
    int lcm = find_LCM(A, B);
  
    for (int i = 1; i <= n; i++)
        cnt += (m + (i % lcm)) / lcm;
  
    return cnt;
}
  
// Driver code
int main()
{
    int n = 60, m = 90, A = 5, B = 10;
  
    cout << CountPairs(n, m, A, B);
  
    return 0;
}

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Java

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//Java implementation of above approach
import java.util.*;
public class ACE {
  
    static int gcd(int a,int b)
    {
        return b==0 ? a :gcd(b,a%b);
    }
      
    //Function to find the LCM
    static int find_LCM(int x, int y)
    {
     return (x * y) / gcd(x, y);
    }
  
    //Function to count the pairs
    static int CountPairs(int n, int m, int A, int B)
    {
     int cnt = 0;
     int lcm = find_LCM(A, B);
  
     for (int i = 1; i <= n; i++)
         cnt += (m + (i % lcm)) / lcm;
  
     return cnt;
    }
  
    //Driver code
    public static void main(String[] args) {
          
        int n = 60, m = 90, A = 5, B = 10;
  
        System.out.println(CountPairs(n, m, A, B));
  
    }
  
}

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Python 3

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# Python3 implementation of
# above approach
  
# from math lib import gcd method
from math import gcd
  
# Function to find the LCM 
def find_LCM(x, y) :
  
    return (x * y) // gcd(x, y)
  
# Function to count the pairs 
def CountPairs(n, m, A, B) :
  
    cnt = 0
    lcm = find_LCM(A, B)
  
    for i in range(1, n + 1) :
        cnt += (m + (i % lcm)) // lcm
  
    return cnt
  
# Driver code     
if __name__ == "__main__" :
  
    n, m, A, B = 60, 90, 5, 10
  
    print(CountPairs(n, m, A, B))
  
# This code is contributed
# by ANKITRAI1

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C#

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// C# implementation of above approach
using System;
  
class GFG
{
    static int gcd(int a,int b)
    {
        return b == 0 ? a : gcd(b, a % b);
    }
      
    // Function to find the LCM
    static int find_LCM(int x, int y)
    {
    return (x * y) / gcd(x, y);
    }
  
    //Function to count the pairs
    static int CountPairs(int n, int m,
                          int A, int B)
    {
        int cnt = 0;
        int lcm = find_LCM(A, B);
      
        for (int i = 1; i <= n; i++)
            cnt += (m + (i % lcm)) / lcm;
      
        return cnt;
    }
  
    // Driver code
    public static void Main() 
    {
        int n = 60, m = 90, A = 5, B = 10;
  
        Console.WriteLine(CountPairs(n, m, A, B));
    }
}
  
// This Code is contributed by mits

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PHP

Output:

540

Time Complexity: O(n)



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