# Count pairs of natural numbers with GCD equal to given number

Given three positive integer **L**, **R**, **G**. The task is to find the count of the pair (x,y) having GCD(x,y) = G and x, y lie between L and R.

Examples:

Input : L = 1, R = 11, G = 5 Output : 3 (5, 5), (5, 10), (10, 5) are three pair having GCD equal to 5 and lie between 1 and 11. So answer is 3. Input : L = 1, R = 10, G = 7 Output : 1

A **simple solution** is to go through all pairs in [L, R]. For every pair, find its GCD. If GCD is equal to g, then increment count. Finally return count.

An **efficient solution** is based on the fact that, for any positive integer pair (x, y) to have GCD equal to g, x and y should be divisible by g.

Observe, there will be at most (R – L)/g numbers between L and R which are divisible by g.

So we find numbers between L and R which are divisible by g. For this, we start from ceil(L/g) * g and with increment by g at each step while it doesn’t exceed R, count numbers having GCD equal to 1.

Also,

ceil(L/g) * g = floor((L + g - 1) / g) * g.

Below is the implementation of above idea :

## C++

`// C++ program to count pair in range of natural ` `// number having GCD equal to given number. ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Return the GCD of two numbers. ` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `return` `b ? gcd(b, a % b) : a; ` `} ` ` ` `// Return the count of pairs having GCD equal to g. ` `int` `countGCD(` `int` `L, ` `int` `R, ` `int` `g) ` `{ ` ` ` `// Setting the value of L, R. ` ` ` `L = (L + g - 1) / g; ` ` ` `R = R/ g; ` ` ` ` ` `// For each possible pair check if GCD is 1. ` ` ` `int` `ans = 0; ` ` ` `for` `(` `int` `i = L; i <= R; i++) ` ` ` `for` `(` `int` `j = L; j <= R; j++) ` ` ` `if` `(gcd(i, j) == 1) ` ` ` `ans++; ` ` ` ` ` `return` `ans; ` `} ` ` ` `// Driven Program ` `int` `main() ` `{ ` ` ` `int` `L = 1, R = 11, g = 5; ` ` ` `cout << countGCD(L, R, g) << endl; ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to count pair in ` `// range of natural number having ` `// GCD equal to given number. ` `import` `java.util.*; ` ` ` `class` `GFG { ` ` ` `// Return the GCD of two numbers. ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `return` `b > ` `0` `? gcd(b, a % b) : a; ` `} ` ` ` `// Return the count of pairs ` `// having GCD equal to g. ` `static` `int` `countGCD(` `int` `L, ` `int` `R, ` `int` `g) { ` ` ` ` ` `// Setting the value of L, R. ` ` ` `L = (L + g - ` `1` `) / g; ` ` ` `R = R / g; ` ` ` ` ` `// For each possible pair check if GCD is 1. ` ` ` `int` `ans = ` `0` `; ` ` ` `for` `(` `int` `i = L; i <= R; i++) ` ` ` `for` `(` `int` `j = L; j <= R; j++) ` ` ` `if` `(gcd(i, j) == ` `1` `) ` ` ` `ans++; ` ` ` ` ` `return` `ans; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main(String[] args) { ` ` ` ` ` `int` `L = ` `1` `, R = ` `11` `, g = ` `5` `; ` ` ` `System.out.println(countGCD(L, R, g)); ` `} ` `} ` ` ` `// This code is contributed by Anant Agarwal. ` |

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## Python3

`# Python program to count ` `# pair in range of natural ` `# number having GCD equal ` `# to given number. ` ` ` `# Return the GCD of two numbers. ` `def` `gcd(a,b): ` ` ` ` ` `return` `gcd(b, a ` `%` `b) ` `if` `b>` `0` `else` `a ` ` ` ` ` `# Return the count of pairs ` `# having GCD equal to g. ` `def` `countGCD(L,R,g): ` ` ` ` ` `# Setting the value of L, R. ` ` ` `L ` `=` `(L ` `+` `g ` `-` `1` `) ` `/` `/` `g ` ` ` `R ` `=` `R` `/` `/` `g ` ` ` ` ` `# For each possible pair ` ` ` `# check if GCD is 1. ` ` ` `ans ` `=` `0` ` ` `for` `i ` `in` `range` `(L,R` `+` `1` `): ` ` ` `for` `j ` `in` `range` `(L,R` `+` `1` `): ` ` ` `if` `(gcd(i, j) ` `=` `=` `1` `): ` ` ` `ans` `=` `ans ` `+` `1` ` ` ` ` `return` `ans ` ` ` `# Driver code ` ` ` `L ` `=` `1` `R ` `=` `11` `g ` `=` `5` ` ` `print` `(countGCD(L, R, g)) ` ` ` `# This code is contributed ` `# by Anant Agarwal. ` |

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## C#

`// C# program to count pair in ` `// range of natural number having ` `// GCD equal to given number. ` `using` `System; ` ` ` `class` `GFG { ` ` ` `// Return the GCD of two numbers. ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `return` `b > 0 ? gcd(b, a % b) : a; ` `} ` ` ` `// Return the count of pairs ` `// having GCD equal to g. ` `static` `int` `countGCD(` `int` `L, ` `int` `R, ` ` ` `int` `g) ` `{ ` ` ` ` ` `// Setting the value of L, R. ` ` ` `L = (L + g - 1) / g; ` ` ` `R = R / g; ` ` ` ` ` `// For each possible pair ` ` ` `// check if GCD is 1. ` ` ` `int` `ans = 0; ` ` ` `for` `(` `int` `i = L; i <= R; i++) ` ` ` `for` `(` `int` `j = L; j <= R; j++) ` ` ` `if` `(gcd(i, j) == 1) ` ` ` `ans++; ` ` ` ` ` `return` `ans; ` `} ` ` ` `// Driver code ` `public` `static` `void` `Main() ` `{ ` ` ` ` ` `int` `L = 1, R = 11, g = 5; ` ` ` `Console.WriteLine(countGCD(L, R, g)); ` `} ` `} ` ` ` `// This code is contributed by vt_m. ` |

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## PHP

`<?php ` `// PHP program to count pair ` `// in range of natural number ` `// having GCD equal to given number. ` ` ` `// Return the GCD of two numbers. ` `function` `gcd( ` `$a` `, ` `$b` `) ` `{ ` ` ` `return` `$b` `? gcd(` `$b` `, ` `$a` `% ` `$b` `) : ` `$a` `; ` `} ` ` ` `// Return the count of pairs ` `// having GCD equal to g. ` `function` `countGCD( ` `$L` `, ` `$R` `, ` `$g` `) ` `{ ` ` ` ` ` `// Setting the value of L, R. ` ` ` `$L` `= (` `$L` `+ ` `$g` `- 1) / ` `$g` `; ` ` ` `$R` `= ` `$R` `/ ` `$g` `; ` ` ` ` ` `// For each possible pair ` ` ` `// check if GCD is 1. ` ` ` `$ans` `= 0; ` ` ` `for` `(` `$i` `= ` `$L` `; ` `$i` `<= ` `$R` `; ` `$i` `++) ` ` ` `for` `(` `$j` `= ` `$L` `; ` `$j` `<= ` `$R` `; ` `$j` `++) ` ` ` `if` `(gcd(` `$i` `, ` `$j` `) == 1) ` ` ` `$ans` `++; ` ` ` ` ` `return` `$ans` `; ` `} ` ` ` ` ` `// Driver Code ` ` ` `$L` `= 1; ` ` ` `$R` `= 11; ` ` ` `$g` `= 5; ` ` ` `echo` `countGCD(` `$L` `, ` `$R` `, ` `$g` `); ` ` ` `// This code is contributed by anuj_67. ` `?> ` |

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Output:

3

This article is contributed by **Anuj Chauhan**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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