Count of all possible pairs having sum of LCM and GCD equal to N

Given an integer N, the task is to find the count of all possible pairs of integers (A, B) such that GCD (A, B) + LCM (A, B) = N.

Examples:

Input: N = 14
Output: 7
Explanation: 
All the possible pairs are {1, 13}, {2, 12}, {4, 6}, {6, 4}, {7, 7}, {12, 2}, {13, 1}

Input: N = 6
Output: 5

Approach:
Follow the steps below to solve the problem:

• Initialize a variable count, to store the count of all the possible pairs.
• Iterate over the range [1, N] to generate all possible pairs (i, j). Calculate the GCD of (i, j) using the __gcd() function and calculate LCM of (i, j).
• Now, check if the sum of LCM (i, j) and GCD (i, j) is equal to N or not. If so, increment count.
• Print the count value after the complete traversal of the range.

Below is the implementation of the above approach:

C++

// C++ Program to implement
// the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to calculate and
// return LCM of two numbers
int lcm(int a, int b)
{
    return (a * b) / __gcd(a, b);
}

// Function to count pairs
// whose sum of GCD and LCM
// is equal to N
int countPair(int N)
{
    int count = 0;
    for (int i = 1;
        i <= N; i++) {

        for (int j = 1;
            j <= N; j++) {

            if (__gcd(i, j)
                    + lcm(i, j)
                == N) {

                count++;
            }
        }
    }

    return count;
}

// Driver Code
int main()
{
    int N = 14;
    cout << countPair(N);

    return 0;
}

Java

// Java program to implement
// the above approach
class GFG{

// Recursive function to return gcd of a and b
static int __gcd(int a, int b)
{
    return b == 0 ? a : __gcd(b, a % b);
}

// Function to calculate and
// return LCM of two numbers
static int lcm(int a, int b)
{
    return (a * b) / __gcd(a, b);
}

// Function to count pairs
// whose sum of GCD and LCM
// is equal to N
static int countPair(int N)
{
    int count = 0;
    for(int i = 1; i <= N; i++)
    {
        for(int j = 1; j <= N; j++)
        {
            if (__gcd(i, j) + lcm(i, j) == N)
            {
                count++;
            }
        }
    }
    return count;
}

// Driver Code
public static void main(String[] args)
{
    int N = 14;

    System.out.print(countPair(N));
}
}

// This code is contributed by Rajput-Ji

Python3

# Python3 program to implement
# the above approach

# Recursive function to return
# gcd of a and b
def __gcd(a, b):

    if b == 0:
        return a
    else:
        return __gcd(b, a % b)

# Function to calculate and
# return LCM of two numbers
def lcm(a, b):

    return (a * b) // __gcd(a, b)

# Function to count pairs
# whose sum of GCD and LCM
# is equal to N
def countPair(N):

    count = 0

    for i in range(1, N + 1):
        for j in range(1, N + 1):
            if (__gcd(i, j) + lcm(i, j) == N):
                count += 1

    return count

# Driver code
if __name__=="__main__":

    N = 14

    print(countPair(N))

# This code is contributed by rutvik_56

C#

// C# program to implement
// the above approach
using System;

class GFG{

// Recursive function to return gcd of a and b
static int __gcd(int a, int b)
{
    return b == 0 ? a : __gcd(b, a % b);
}

// Function to calculate and
// return LCM of two numbers
static int lcm(int a, int b)
{
    return (a * b) / __gcd(a, b);
}

// Function to count pairs
// whose sum of GCD and LCM
// is equal to N
static int countPair(int N)
{
    int count = 0;
    for(int i = 1; i <= N; i++)
    {
        for(int j = 1; j <= N; j++)
        {
            if (__gcd(i, j) + lcm(i, j) == N)
            {
                count++;
            }
        }
    }
    return count;
}

// Driver Code
public static void Main(String[] args)
{
    int N = 14;

    Console.Write(countPair(N));
}
}

// This code is contributed by gauravrajput1

Javascript

<script>
// javascript program to implement
// the above approach
// Recursive function to return gcd of a and b
    function __gcd(a , b)
    {
        return b == 0 ? a : __gcd(b, a % b);
    }

    // Function to calculate and
    // return LCM of two numbers
    function lcm(a , b) {
        return (a * b) / __gcd(a, b);
    }

    // Function to count pairs
    // whose sum of GCD and LCM
    // is equal to N
    function countPair(N)
    {
        var count = 0;
        for (i = 1; i <= N; i++) {
            for (j = 1; j <= N; j++) {
                if (__gcd(i, j) + lcm(i, j) == N) {
                    count++;
                }
            }
        }
        return count;
    }

    // Driver Code
        var N = 14;
        document.write(countPair(N));

// This code is contributed by aashish1995
</script>

Output:

7

Time Complexity: O(N²*log(N)) (here two nested for loops so O(n²) and in between them there is __gcd(a,b) which will run in log(N) time so effective time complexity will be O(N²*log(N)) )
Auxiliary Space: O(logn)