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Carmichael Numbers
• Difficulty Level : Easy
• Last Updated : 14 May, 2018

A number n is said to be a Carmichael number if it satisfies the following modular arithmetic condition:

```  power(b, n-1) MOD n = 1,
for all b ranging from 1 to n such that b and
n are relatively prime, i.e, gcd(b, n) = 1 ```

Given a positive integer n, find if it is a Carmichael number. These numbers have importance in Fermat Method for primality testing.

Examples :

```Input :  n = 8
Output : false
Explanation : 8 is not a Carmichael number because 3 is
relatively prime to 8 and (38-1) % 8
= 2187 % 8 is not 1.

Input :  n = 561
Output : true
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

The idea is simple, we iterate through all numbers from 1 to n and for every relatively prime number, we check if its (n-1)th power under modulo n is 1 or not.

Below is a the program to check if a given number is Carmichael or not.

## C++

 `// A C++ program to check if a number is``// Carmichael or not.``#include ``using` `namespace` `std;`` ` `// utility function to find gcd of two numbers``int` `gcd(``int` `a, ``int` `b)``{``    ``if` `(a < b)``        ``return` `gcd(b, a);``    ``if` `(a % b == 0)``        ``return` `b;``    ``return` `gcd(b, a % b);``}`` ` `// utility function to find pow(x, y) under``// given modulo mod``int` `power(``int` `x, ``int` `y, ``int` `mod)``{``    ``if` `(y == 0)``        ``return` `1;``    ``int` `temp = power(x, y / 2, mod) % mod;``    ``temp = (temp * temp) % mod;``    ``if` `(y % 2 == 1)``        ``temp = (temp * x) % mod;``    ``return` `temp;``}`` ` `// This function receives an integer n and``// finds if it's a Carmichael number``bool` `isCarmichaelNumber(``int` `n)``{``    ``for` `(``int` `b = 2; b < n; b++) {``        ``// If "b" is relatively prime to n``        ``if` `(gcd(b, n) == 1)`` ` `            ``// And pow(b, n-1)%n is not 1,``            ``// return false.``            ``if` `(power(b, n - 1, n) != 1)``                ``return` `false``;``    ``}``    ``return` `true``;``}`` ` `// Driver function``int` `main()``{``    ``cout << isCarmichaelNumber(500) << endl;``    ``cout << isCarmichaelNumber(561) << endl;``    ``cout << isCarmichaelNumber(1105) << endl;``    ``return` `0;``}`

## Java

 `// JAVA program to check if a number is``// Carmichael or not.``import` `java.io.*;`` ` `class` `GFG {`` ` `    ``// utility function to find gcd of``    ``// two numbers``    ``static` `int` `gcd(``int` `a, ``int` `b)``    ``{``        ``if` `(a < b)``            ``return` `gcd(b, a);``        ``if` `(a % b == ``0``)``            ``return` `b;``        ``return` `gcd(b, a % b);``    ``}`` ` `    ``// utility function to find pow(x, y)``    ``// under given modulo mod``    ``static` `int` `power(``int` `x, ``int` `y, ``int` `mod)``    ``{``        ``if` `(y == ``0``)``            ``return` `1``;``        ``int` `temp = power(x, y / ``2``, mod) % mod;``        ``temp = (temp * temp) % mod;``        ``if` `(y % ``2` `== ``1``)``            ``temp = (temp * x) % mod;``        ``return` `temp;``    ``}`` ` `    ``// This function receives an integer n and``    ``// finds if it's a Carmichael number``    ``static` `int` `isCarmichaelNumber(``int` `n)``    ``{``        ``for` `(``int` `b = ``2``; b < n; b++) {``            ``// If "b" is relatively prime to n``            ``if` `(gcd(b, n) == ``1``)`` ` `                ``// And pow(b, n-1)%n is not 1,``                ``// return false.``                ``if` `(power(b, n - ``1``, n) != ``1``)``                    ``return` `0``;``        ``}``        ``return` `1``;``    ``}`` ` `    ``// Driver function``    ``public` `static` `void` `main(String args[])``    ``{``        ``System.out.println(isCarmichaelNumber(``500``));``        ``System.out.println(isCarmichaelNumber(``561``));``        ``System.out.println(isCarmichaelNumber(``1105``));``    ``}``}``// This code is contributed by Nikita Tiwari.`

## Python

 `# A Python program to check if a number is``# Carmichael or not.`` ` `# utility function to find gcd of two numbers``def` `gcd( a, b) :``    ``if` `(a < b) :``        ``return` `gcd(b, a)``    ``if` `(a ``%` `b ``=``=` `0``) :``        ``return` `b``    ``return` `gcd(b, a ``%` `b)`` ` `# utility function to find pow(x, y) under``# given modulo mod``def` `power(x, y, mod) :``    ``if` `(y ``=``=` `0``) :``        ``return` `1``    ``temp ``=` `power(x, y ``/` `2``, mod) ``%` `mod``    ``temp ``=` `(temp ``*` `temp) ``%` `mod``    ``if` `(y ``%` `2` `=``=` `1``) :``        ``temp ``=` `(temp ``*` `x) ``%` `mod``    ``return` `temp`` ` ` ` `# This function receives an integer n and``# finds if it's a Carmichael number``def` `isCarmichaelNumber( n) :``    ``b ``=` `2``    ``while` `b

## C#

 `// C# program to check if a number is``// Carmichael or not.``using` `System;`` ` `class` `GFG {`` ` `    ``// utility function to find gcd of``    ``// two numbers``    ``static` `int` `gcd(``int` `a, ``int` `b)``    ``{``        ``if` `(a < b)``            ``return` `gcd(b, a);``        ``if` `(a % b == 0)``            ``return` `b;``        ``return` `gcd(b, a % b);``    ``}`` ` `    ``// utility function to find pow(x, y)``    ``// under given modulo mod``    ``static` `int` `power(``int` `x, ``int` `y, ``int` `mod)``    ``{``        ``if` `(y == 0)``            ``return` `1;`` ` `        ``int` `temp = power(x, y / 2, mod) % mod;``        ``temp = (temp * temp) % mod;`` ` `        ``if` `(y % 2 == 1)``            ``temp = (temp * x) % mod;`` ` `        ``return` `temp;``    ``}`` ` `    ``// This function receives an integer n and``    ``// finds if it's a Carmichael number``    ``static` `int` `isCarmichaelNumber(``int` `n)``    ``{``        ``for` `(``int` `b = 2; b < n; b++) {``            ``// If "b" is relatively prime to n``            ``if` `(gcd(b, n) == 1)`` ` `                ``// And pow(b, n-1)%n is not 1,``                ``// return false.``                ``if` `(power(b, n - 1, n) != 1)``                    ``return` `0;``        ``}``        ``return` `1;``    ``}`` ` `    ``// Driver function``    ``public` `static` `void` `Main()``    ``{``        ``Console.WriteLine(isCarmichaelNumber(500));``        ``Console.WriteLine(isCarmichaelNumber(561));``        ``Console.WriteLine(isCarmichaelNumber(1105));``    ``}``}`` ` `// This code is contributed by vt_m.`

## PHP

 ``

Output :
```0
1
1
```

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