# 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

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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;` `}` |

## 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);` `}` `}` |

## 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` |

## 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` |

## 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` `?>` |

## Javascript

`<script>` `// A simple Javascript 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` ` ` `document.write(` `"1 "` `);` ` ` `for` `(` `var` `i = 2; i < n; i++)` ` ` `// A number x is generator of` ` ` `// GCD is 1` ` ` `if` `(gcd(i, n) == 1)` ` ` `document.write(i + ` `" "` `);` `}` `// Driver Code` `var` `n = 10;` `printGenerators(n);` `// This code is contributed by Kirti` `</script>` |

**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.

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