# 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   ` `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  `

## 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 `

Output:

```7
```

Time Complexity: O(N3)
Auxiliary Space: O(1)

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Improved By : Rajput-Ji, GauravRajput1