# Count of numbers upto M with GCD equals to K when paired with M

Given two integers M and K, the task is to count the number of integers between [0, M] such that GCD of that integer with M equals to K.

Examples:

Input: M = 9, K = 1
Output: 6
Explanation:
The possible numbers such that when paired with 9, there GCD is 1, are 1, 2, 4, 5, 7, 8.

Input: M = 10, K = 5
Output: 1

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

• Integers having GCD K with M will be of the form K, 2K, 3K, …..and so on up to M.
• Let’s consider the coefficients of K i.e 1, 2, 3, 4…up to (M/K).
• Now we just have to find the count of such coefficients which have GCD with the number (M/K) = 1. So now problem reduces to find the number of integers between 1 to (M/K) having Gcd with (m/k) = 1.
• To find this we will use the Euler totient function of (M/K).

Below is the implementation of the above approach:

## C++

 `// C++ program to Count of numbers ` `// between 0 to M which have GCD ` `// with M equals to K. ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to calculate GCD ` `// using euler totient function ` `int` `EulerTotientFunction(``int` `limit) ` `{ ` `    ``int` `copy = limit; ` ` `  `    ``// Finding the prime factors of ` `    ``// limit to calculate it's ` `    ``// euler totient function ` `    ``vector<``int``> primes; ` ` `  `    ``for` `(``int` `i = 2; i * i <= limit; i++) { ` `        ``if` `(limit % i == 0) { ` `            ``while` `(limit % i == 0) { ` `                ``limit /= i; ` `            ``} ` `            ``primes.push_back(i); ` `        ``} ` `    ``} ` `    ``if` `(limit >= 2) { ` `        ``primes.push_back(limit); ` `    ``} ` ` `  `    ``// Calculating the euler totien ` `    ``// function of (m/k) ` `    ``int` `ans = copy; ` `    ``for` `(``auto` `it : primes) { ` `        ``ans = (ans / it) * (it - 1); ` `    ``} ` `    ``return` `ans; ` `} ` ` `  `// Function print the count of ` `// numbers whose GCD with M ` `// equals to K ` `void` `CountGCD(``int` `m, ``int` `k) ` `{ ` ` `  `    ``if` `(m % k != 0) { ` `        ``// GCD of m with any integer ` `        ``// cannot  be equal to k ` `        ``cout << 0 << endl; ` `        ``return``; ` `    ``} ` ` `  `    ``if` `(m == k) { ` `        ``// 0 and m itself will be ` `        ``// the only valid integers ` `        ``cout << 2 << endl; ` `        ``return``; ` `    ``} ` ` `  `    ``// Finding the number upto which ` `    ``// coefficient of k can come ` `    ``int` `limit = m / k; ` ` `  `    ``int` `ans = EulerTotientFunction(limit); ` ` `  `    ``cout << ans << endl; ` `} ` `// Driver code ` `int` `main() ` `{ ` ` `  `    ``int` `M = 9; ` `    ``int` `K = 1; ` `    ``CountGCD(M, K); ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java program to Count of numbers ` `// between 0 to M which have GCD ` `// with M equals to K. ` `import` `java.util.*; ` ` `  `class` `GFG{ ` `  `  `// Function to calculate GCD ` `// using euler totient function ` `static` `int` `EulerTotientFunction(``int` `limit) ` `{ ` `    ``int` `copy = limit; ` `  `  `    ``// Finding the prime factors of ` `    ``// limit to calculate it's ` `    ``// euler totient function ` `    ``Vector primes = ``new` `Vector(); ` `  `  `    ``for` `(``int` `i = ``2``; i * i <= limit; i++) { ` `        ``if` `(limit % i == ``0``) { ` `            ``while` `(limit % i == ``0``) { ` `                ``limit /= i; ` `            ``} ` `            ``primes.add(i); ` `        ``} ` `    ``} ` `    ``if` `(limit >= ``2``) { ` `        ``primes.add(limit); ` `    ``} ` `  `  `    ``// Calculating the euler totien ` `    ``// function of (m/k) ` `    ``int` `ans = copy; ` `    ``for` `(``int` `it : primes) { ` `        ``ans = (ans / it) * (it - ``1``); ` `    ``} ` `    ``return` `ans; ` `} ` `  `  `// Function print the count of ` `// numbers whose GCD with M ` `// equals to K ` `static` `void` `CountGCD(``int` `m, ``int` `k) ` `{ ` `  `  `    ``if` `(m % k != ``0``) { ` `        ``// GCD of m with any integer ` `        ``// cannot  be equal to k ` `        ``System.out.print(``0` `+``"\n"``); ` `        ``return``; ` `    ``} ` `  `  `    ``if` `(m == k) { ` `        ``// 0 and m itself will be ` `        ``// the only valid integers ` `        ``System.out.print(``2` `+``"\n"``); ` `        ``return``; ` `    ``} ` `  `  `    ``// Finding the number upto which ` `    ``// coefficient of k can come ` `    ``int` `limit = m / k; ` `  `  `    ``int` `ans = EulerTotientFunction(limit); ` `  `  `    ``System.out.print(ans +``"\n"``); ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` `  `  `    ``int` `M = ``9``; ` `    ``int` `K = ``1``; ` `    ``CountGCD(M, K); ` `  `  `} ` `} ` ` `  `// This code is contributed by sapnasingh4991 `

## Python3

 `# Python3 program to Count of numbers ` `# between 0 to M which have GCD ` `# with M equals to K. ` ` `  `# Function to calculate GCD ` `# using euler totient function ` `def` `EulerTotientFunction(limit): ` `    ``copy ``=` `limit ` ` `  `    ``# Finding the prime factors of ` `    ``# limit to calculate it's ` `    ``# euler totient function ` `    ``primes ``=` `[] ` ` `  `    ``for` `i ``in` `range``(``2``, limit ``+` `1``): ` `        ``if` `i ``*` `i > limit: ` `            ``break` `        ``if` `(limit ``%` `i ``=``=` `0``): ` `            ``while` `(limit ``%` `i ``=``=` `0``): ` `                ``limit ``/``/``=` `i ` `            ``primes.append(i) ` ` `  `    ``if` `(limit >``=` `2``): ` `        ``primes.append(limit) ` ` `  `    ``# Calculating the euler totien ` `    ``# function of (m//k) ` `    ``ans ``=` `copy ` `    ``for` `it ``in` `primes: ` `        ``ans ``=` `(ans ``/``/` `it) ``*` `(it ``-` `1``) ` ` `  `    ``return` `ans ` ` `  `# Function print the count of ` `# numbers whose GCD with M ` `# equals to K ` `def` `CountGCD(m, k): ` ` `  `    ``if` `(m ``%` `k !``=` `0``): ` `         `  `        ``# GCD of m with any integer ` `        ``# cannot be equal to k ` `        ``print``(``0``) ` `        ``return` ` `  `    ``if` `(m ``=``=` `k): ` `         `  `        ``# 0 and m itself will be ` `        ``# the only valid integers ` `        ``print``(``2``) ` `        ``return` ` `  `    ``# Finding the number upto which ` `    ``# coefficient of k can come ` `    ``limit ``=` `m ``/``/` `k ` ` `  `    ``ans ``=` `EulerTotientFunction(limit) ` ` `  `    ``print``(ans) ` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` ` `  `    ``M ``=` `9` `    ``K ``=` `1` `    ``CountGCD(M, K) ` ` `  `# This code is contributed by mohit kumar 29     `

## C#

 `// C# program to Count of numbers ` `// between 0 to M which have GCD ` `// with M equals to K. ` `using` `System; ` `using` `System.Collections.Generic; ` ` `  `class` `GFG{ ` ` `  `// Function to calculate GCD ` `// using euler totient function ` `static` `int` `EulerTotientFunction(``int` `limit) ` `{ ` `    ``int` `copy = limit; ` ` `  `    ``// Finding the prime factors of ` `    ``// limit to calculate it's ` `    ``// euler totient function ` `    ``List<``int``> primes = ``new` `List<``int``>(); ` ` `  `    ``for` `(``int` `i = 2; i * i <= limit; i++)  ` `    ``{ ` `        ``if` `(limit % i == 0)  ` `        ``{ ` `            ``while` `(limit % i == 0)  ` `            ``{ ` `                ``limit /= i; ` `            ``} ` `            ``primes.Add(i); ` `        ``} ` `    ``} ` `    ``if` `(limit >= 2)  ` `    ``{ ` `        ``primes.Add(limit); ` `    ``} ` ` `  `    ``// Calculating the euler totien ` `    ``// function of (m/k) ` `    ``int` `ans = copy; ` `    ``foreach` `(``int` `it ``in` `primes)  ` `    ``{ ` `        ``ans = (ans / it) * (it - 1); ` `    ``} ` `    ``return` `ans; ` `} ` ` `  `// Function print the count of ` `// numbers whose GCD with M ` `// equals to K ` `static` `void` `CountGCD(``int` `m, ``int` `k) ` `{ ` `    ``if` `(m % k != 0)  ` `    ``{ ` `        ``// GCD of m with any integer ` `        ``// cannot be equal to k ` `        ``Console.Write(0 + ``"\n"``); ` `        ``return``; ` `    ``} ` ` `  `    ``if` `(m == k)  ` `    ``{ ` `        ``// 0 and m itself will be ` `        ``// the only valid integers ` `        ``Console.Write(2 + ``"\n"``); ` `        ``return``; ` `    ``} ` ` `  `    ``// Finding the number upto which ` `    ``// coefficient of k can come ` `    ``int` `limit = m / k; ` ` `  `    ``int` `ans = EulerTotientFunction(limit); ` ` `  `    ``Console.Write(ans + ``"\n"``); ` `} ` ` `  `// Driver code ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `M = 9; ` `    ``int` `K = 1; ` `    ``CountGCD(M, K); ` `} ` `} ` ` `  `// This code is contributed by PrinciRaj1992 `

Output:

```6
```

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

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.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :
Practice Tags :

Be the First to upvote.

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.