# Number Theory | Generators of finite cyclic group under addition

Given a number n, find all generators of cyclic additive group under modulo n. Generator of a set {0, 1, … n-1} is an element x such that x is smaller than n, and using x (and addition operation), we can generate all elements of the set.

**Examples:**

Input : 10 Output : 1 3 7 9 The set to be generated is {0, 1, .. 9} By adding 1, single or more times, we can create all elements from 0 to 9. Similarly using 3, we can generate all elements. 30 % 10 = 0, 21 % 10 = 1, 12 % 10 = 2, ... Same is true for 7 and 9. Input : 24 Output : 1 5 7 11 13 17 19 23

A **simple solution** is to run a loop from 1 to n-1 and for every element check if it is generator. To check generator, we keep adding element and we check if we can generate all numbers until remainder starts repeating.

An **Efficient solution** is based on the fact that a number x is generator if x is relatively prime to n, i.e., gcd(n, x) =1.

Below is the implementation of above approach:

## C++

`// A simple C++ program to find all generators ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to return gcd of a and b ` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(a == 0) ` ` ` `return` `b; ` ` ` `return` `gcd(b%a, a); ` `} ` ` ` `// Print generators of n ` `int` `printGenerators(unsigned ` `int` `n) ` `{ ` ` ` `// 1 is always a generator ` ` ` `cout << ` `"1 "` `; ` ` ` ` ` `for` `(` `int` `i=2; i < n; i++) ` ` ` ` ` `// A number x is generator of GCD is 1 ` ` ` `if` `(gcd(i, n) == 1) ` ` ` `cout << i << ` `" "` `; ` `} ` ` ` `// Driver program to test above function ` `int` `main() ` `{ ` ` ` `int` `n = 10; ` ` ` `printGenerators(n); ` ` ` `return` `0; ` `} ` |

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## Java

`// A simple Java program to find all generators ` ` ` `class` `GFG { ` ` ` ` ` `// Function to return gcd of a and b ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(a == ` `0` `) ` ` ` `return` `b; ` ` ` `return` `gcd(b%a, a); ` `} ` ` ` `// Print generators of n ` `static` `void` `printGenerators(` `int` `n) ` `{ ` ` ` `// 1 is always a generator ` ` ` `System.out.println(` `"1 "` `); ` ` ` ` ` `for` `(` `int` `i=` `2` `; i < n; i++) ` ` ` ` ` `// A number x is generator of GCD is 1 ` ` ` `if` `(gcd(i, n) == ` `1` `) ` ` ` `System.out.println(i +` `" "` `); ` `} ` ` ` `// Driver program to test above function ` `public` `static` `void` `main(String args[]) ` `{ ` ` ` `int` `n = ` `10` `; ` ` ` `printGenerators(n); ` `} ` `} ` |

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## Python3

`# Python3 program to find all generators ` ` ` `# Function to return gcd of a and b ` `def` `gcd(a, b): ` ` ` `if` `(a ` `=` `=` `0` `): ` ` ` `return` `b; ` ` ` `return` `gcd(b ` `%` `a, a); ` ` ` `# Print generators of n ` `def` `printGenerators(n): ` ` ` ` ` `# 1 is always a generator ` ` ` `print` `(` `"1"` `, end ` `=` `" "` `); ` ` ` ` ` `for` `i ` `in` `range` `(` `2` `, n): ` ` ` ` ` `# A number x is generator ` ` ` `# of GCD is 1 ` ` ` `if` `(gcd(i, n) ` `=` `=` `1` `): ` ` ` `print` `(i, end ` `=` `" "` `); ` ` ` `# Driver Code ` `n ` `=` `10` `; ` `printGenerators(n); ` ` ` `# This code is contributed by mits ` |

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## C#

`// A simple C# program to find all generators ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` `// Function to return gcd of a and b ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(a == 0) ` ` ` `return` `b; ` ` ` `return` `gcd(b % a, a); ` `} ` ` ` `// Print generators of n ` `static` `void` `printGenerators(` `int` `n) ` `{ ` ` ` `// 1 is always a generator ` ` ` `Console.Write(` `"1 "` `); ` ` ` ` ` `for` `(` `int` `i = 2; i < n; i++) ` ` ` ` ` `// A number x is generator of GCD is 1 ` ` ` `if` `(gcd(i, n) == 1) ` ` ` `Console.Write(i +` `" "` `); ` `} ` ` ` `// Driver code ` `public` `static` `void` `Main(String []args) ` `{ ` ` ` `int` `n = 10; ` ` ` `printGenerators(n); ` `} ` `} ` ` ` `// This code contributed by Rajput-Ji ` |

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## PHP

`<?php ` `// PHP program to find all generators ` ` ` `// Function to return gcd of a and b ` ` ` `function` `gcd(` `$a` `, ` `$b` `) ` `{ ` ` ` `if` `(` `$a` `== 0) ` ` ` `return` `$b` `; ` ` ` `return` `gcd(` `$b` `% ` `$a` `, ` `$a` `); ` `} ` ` ` `// Print generators of n ` `function` `printGenerators(` `$n` `) ` `{ ` ` ` ` ` `// 1 is always a generator ` ` ` `echo` `"1 "` `; ` ` ` ` ` `for` `(` `$i` `= 2; ` `$i` `< ` `$n` `; ` `$i` `++) ` ` ` ` ` `// A number x is generator ` ` ` `// of GCD is 1 ` ` ` `if` `(gcd(` `$i` `, ` `$n` `) == 1) ` ` ` `echo` `$i` `, ` `" "` `; ` `} ` ` ` `// Driver program to test ` `// above function ` ` ` `$n` `= 10; ` ` ` `printGenerators(` `$n` `); ` ` ` `// This code is contributed by Ajit ` `?> ` |

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**Output :**

1 3 7 9

**How does this work?**

If we consider all remainders of n consecutive multiples of x, then some remainders would repeat if GCD of x and n is not 1. If some remainders repeat, then x cannot be a generator. Note that after n consecutive multiples, remainders would anyway repeat.

**Interesting Observation : **

Number of generators of a number n is equal to Φ(n) where Φ is Euler Totient Function.

This article is contributed by **Ujjwal Goyal**. 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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