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# Euler’s Totient Function

Euler’s Totient function Φ (n) for an input n is the count of numbers in {1, 2, 3, …, n-1} that are relatively prime to n, i.e., the numbers whose GCD (Greatest Common Divisor) with n is 1.

Examples :

Φ(1) = 1
gcd(1, 1) is 1

Φ(2) = 1
gcd(1, 2) is 1, but gcd(2, 2) is 2.

Φ(3) = 2
gcd(1, 3) is 1 and gcd(2, 3) is 1

Φ(4) = 2
gcd(1, 4) is 1 and gcd(3, 4) is 1

Φ(5) = 4
gcd(1, 5) is 1, gcd(2, 5) is 1,
gcd(3, 5) is 1 and gcd(4, 5) is 1

Φ(6) = 2
gcd(1, 6) is 1 and gcd(5, 6) is 1, 
Recommended Practice

How to compute Φ(n) for an input nΦ
A simple solution is to iterate through all numbers from 1 to n-1 and count numbers with gcd with n as 1. Below is the implementation of the simple method to compute Euler’s Totient function for an input integer n.

## C++

 // A simple C++ program to calculate // Euler's Totient Function  #include  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);  }    // A simple method to evaluate Euler Totient Function  int phi(unsigned int n)  {      unsigned int result = 1;      for (int i = 2; i < n; i++)          if (gcd(i, n) == 1)              result++;      return result;  }    // Driver program to test above function  int main()  {      int n;      for (n = 1; n <= 10; n++)          cout << "phi("<

## C

 // A simple C program to calculate Euler's Totient Function #include    // Function to return gcd of a and b int gcd(int a, int b) {     if (a == 0)         return b;     return gcd(b % a, a); }   // A simple method to evaluate Euler Totient Function int phi(unsigned int n) {     unsigned int result = 1;     for (int i = 2; i < n; i++)         if (gcd(i, n) == 1)             result++;     return result; }   // Driver program to test above function int main() {     int n;     for (n = 1; n <= 10; n++)         printf("phi(%d) = %d\n", n, phi(n));     return 0; }

## Java

 // A simple java program to calculate // Euler's Totient Function import java.io.*;   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);     }       // A simple method to evaluate     // Euler Totient Function     static int phi(int n)     {         int result = 1;         for (int i = 2; i < n; i++)             if (gcd(i, n) == 1)                 result++;         return result;     }       // Driver code     public static void main(String[] args)     {         int n;           for (n = 1; n <= 10; n++)             System.out.println("phi(" + n + ") = " + phi(n));     } }   // This code is contributed by sunnusingh

## Python3

 # A simple Python3 program  # to calculate Euler's  # Totient Function   # Function to return # gcd of a and b def gcd(a, b):       if (a == 0):         return b     return gcd(b % a, a)   # A simple method to evaluate # Euler Totient Function def phi(n):       result = 1     for i in range(2, n):         if (gcd(i, n) == 1):             result+=1     return result   # Driver Code for n in range(1, 11):     print("phi(",n,") = ",             phi(n), sep = "")              # This code is contributed # by Smitha

## C#

 // A simple C# program to calculate // Euler's Totient Function 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);     }       // A simple method to evaluate     // Euler Totient Function     static int phi(int n)     {         int result = 1;         for (int i = 2; i < n; i++)             if (gcd(i, n) == 1)                 result++;         return result;     }       // Driver code     public static void Main()     {         for (int n = 1; n <= 10; n++)         Console.WriteLine("phi(" + n + ") = " + phi(n));     } }   // This code is contributed by nitin mittal

## PHP

 <Φphp // PHP program to calculate  // Euler's Totient Function   // Function to return  // gcd of a and b function gcd($a, $b) {     if ($a == 0)  return $b;     return gcd($b % $a, $a); } // A simple method to evaluate // Euler Totient Function function phi($n) {     $result = 1;  for ($i = 2; $i < $n; $i++)  if (gcd($i, $n) == 1)  $result++;     return $result; } // Driver Code for ($n = 1; $n <= 10; $n++)     echo "phi(" .$n. ") =" . phi($n)."\n";   // This code is contributed by Sam007 Φ>

## Javascript

 

Output

phi(1) = 1
phi(2) = 1
phi(3) = 2
phi(4) = 2
phi(5) = 4
phi(6) = 2
phi(7) = 6
phi(8) = 4
phi(9) = 6
phi(10) = 4

The above code calls gcd function O(n) times. The time complexity of the gcd function is O(h) where “h” is the number of digits in a smaller number of given two numbers. Therefore, an upper bound on the time complexity of the above solution is O(N^2 log N) [HowΦ there can be at most Log10n digits in all numbers from 1 to n]

Auxiliary Space: O(log N)

Below is a Better Solution. The idea is based on Euler’s product formula which states that the value of totient functions is below the product overall prime factors p of n. The formula basically says that the value of Φ(n) is equal to n multiplied by-product of (1 – 1/p) for all prime factors p of n. For example value of Φ(6) = 6 * (1-1/2) * (1 – 1/3) = 2.
We can find all prime factors using the idea used in this post.

1) Initialize : result = n
2) Run a loop from 'p' = 2 to sqrt(n), do following for every 'p'.
a) If p divides n, then
Set: result = result  * (1.0 - (1.0 / (float) p));
Divide all occurrences of p in n.
3) Return result  

Below is the implementation of Euler’s product formula.

## C++

 // C++ program to calculate Euler's  // Totient Function using Euler's // product formula #include  using namespace std;   int phi(int n) {           // Initialize result as n     float result = n;         // Consider all prime factors of n      // and for every prime factor p,     // multiply result with (1 - 1/p)     for(int p = 2; p * p <= n; ++p)     {                   // Check if p is a prime factor.         if (n % p == 0)         {                           // If yes, then update n and result             while (n % p == 0)                 n /= p;                               result *= (1.0 - (1.0 / (float)p));         }     }        // If n has a prime factor greater than sqrt(n)     // (There can be at-most one such prime factor)     if (n > 1)         result -= result / n;   //Since in the set {1,2,....,n-1}, all numbers are relatively prime with n   //if n is a prime number        return (int)result; }    // Driver code int main() {     int n;           for(n = 1; n <= 10; n++)     {         cout << "Phi" << "("              << n << ")" << " = "              << phi(n) <

## C

 // C program to calculate Euler's Totient Function // using Euler's product formula #include    int phi(int n) {     float result = n; // Initialize result as n       // Consider all prime factors of n and for every prime     // factor p, multiply result with (1 - 1/p)     for (int p = 2; p * p <= n; ++p) {                   // Check if p is a prime factor.         if (n % p == 0) {                           // If yes, then update n and result             while (n % p == 0)                 n /= p;             result *= (1.0 - (1.0 / (float)p));         }     }       // If n has a prime factor greater than sqrt(n)     // (There can be at-most one such prime factor)     if (n > 1)         result -= result / n;   //Since in the set {1,2,....,n-1}, all numbers are relatively prime with n   //if n is a prime number       return (int)result; }   // Driver program to test above function int main() {     int n;     for (n = 1; n <= 10; n++)         printf("phi(%d) = %d\n", n, phi(n));     return 0; }

## Java

 // Java program to calculate Euler's Totient // Function using Euler's product formula import java.io.*;   class GFG {     static int phi(int n)     {         // Initialize result as n         float result = n;           // Consider all prime factors of n and for         // every prime factor p, multiply result         // with (1 - 1/p)         for (int p = 2; p * p <= n; ++p) {             // Check if p is a prime factor.             if (n % p == 0) {                 // If yes, then update n and result                 while (n % p == 0)                     n /= p;                 result *= (1.0 - (1.0 / (float)p));             }         }           // If n has a prime factor greater than sqrt(n)         // (There can be at-most one such prime factor)         if (n > 1)             result -= result / n;   //Since in the set {1,2,....,n-1}, all numbers are relatively prime with n   //if n is a prime number           return (int)result;     }       // Driver program to test above function     public static void main(String args[])     {         int n;         for (n = 1; n <= 10; n++)             System.out.println("phi(" + n + ") = " + phi(n));     } }   // This code is contributed by Nikita Tiwari.

## Python3

 # Python 3 program to calculate # Euler's Totient Function # using Euler's product formula   def phi(n) :       result = n   # Initialize result as n            # Consider all prime factors     # of n and for every prime     # factor p, multiply result with (1 - 1 / p)     p = 2     while p * p<= n :           # Check if p is a prime factor.         if n % p == 0 :               # If yes, then update n and result             while n % p == 0 :                 n = n // p             result = result * (1.0 - (1.0 / float(p)))         p = p + 1                         # If n has a prime factor     # greater than sqrt(n)     # (There can be at-most one     # such prime factor)     if n > 1 :         result -= result // n   #Since in the set {1,2,....,n-1}, all numbers are relatively prime with n   #if n is a prime number        return int(result)             # Driver program to test above function for n in range(1, 11) :     print("phi(", n, ") = ", phi(n))        # This code is contributed # by Nikita Tiwari.

## C#

 // C# program to calculate Euler's Totient // Function using Euler's product formula using System;   class GFG {           static int phi(int n)     {                   // Initialize result as n         float result = n;           // Consider all prime factors         // of n and for every prime          // factor p, multiply result         // with (1 - 1 / p)         for (int p = 2; p * p <= n; ++p)          {                           // Check if p is a prime factor.             if (n % p == 0)              {                                   // If yes, then update                 // n and result                 while (n % p == 0)                     n /= p;                 result *= (float)(1.0 - (1.0 / (float)p));             }         }           // If n has a prime factor          // greater than sqrt(n)         // (There can be at-most          // one such prime factor)         if (n > 1)             result -= result / n;   //Since in the set {1,2,....,n-1}, all numbers are relatively prime with n   //if n is a prime number           return (int)result;     }       // Driver Code     public static void Main()     {         int n;         for (n = 1; n <= 10; n++)             Console.WriteLine("phi(" + n + ") = " + phi(n));     } }   // This code is contributed by nitin mittal.

## PHP

 <Φphp // PHP program to calculate  // Euler's Totient Function  // using Euler's product formula function phi($n) {  // Initialize result as n  $result = $n;   // Consider all prime factors  // of n and for every prime  // factor p, multiply result   // with (1 - 1/p)  for ($p = 2; $p * $p <= $n; ++$p)      {                   // Check if p is         // a prime factor.         if ($n % $p == 0)          {                           // If yes, then update             // n and result             while ($n % $p == 0)                 $n /= $p;             $result *= (1.0 - (1.0 / $p));         }     }       // If n has a prime factor greater      // than sqrt(n) (There can be at-most     // one such prime factor)     if ($n > 1)  $result -= $result / $n;   //Since in the set {1,2,....,n-1}, all numbers are relatively prime with n   //if n is a prime number       return intval($result); } // Driver Code for ($n = 1; $n <= 10; $n++) echo "phi(" .$n. ") =" . phi($n)."\n";       // This code is contributed by Sam007 Φ>

## Javascript

 // Javascript program to calculate  // Euler's Totient Function  // using Euler's product formula function phi(n) {     // Initialize result as n     let result = n;        // Consider all prime factors     // of n and for every prime     // factor p, multiply result      // with (1 - 1/p)     for (let p = 2; p * p <= n; ++p)      {                   // Check if p is         // a prime factor.         if (n % p == 0)          {                           // If yes, then update             // n and result             while (n % p == 0)                 n /= p;             result *= (1.0 - (1.0 / p));         }     }       // If n has a prime factor greater      // than sqrt(n) (There can be at-most     // one such prime factor)     if (n > 1)         result -= result / n;   //Since in the set {1,2,....,n-1}, all numbers are relatively prime with n   //if n is a prime number       return parseInt(result); }   // Driver Code for (let n = 1; n <= 10; n++)  document.write(phi(${n}) =${phi(n)}
);       // This code is contributed by _saurabh_jaiswal

Output

Phi(1) = 1
Phi(2) = 1
Phi(3) = 2
Phi(4) = 2
Phi(5) = 4
Phi(6) = 2
Phi(7) = 6
Phi(8) = 4
Phi(9) = 6
Phi(10) = 4

Time Complexity: O(√n log n)
Auxiliary Space: O(1)

We can avoid floating-point calculations in the above method. The idea is to count all prime factors and their multiples and subtract this count from n to get the totient function value (Prime factors and multiples of prime factors won’t have gcd as 1)

1) Initialize result as n
2) Consider every number 'p' (where 'p' varies from 2 to Φn).
If p divides n, then do following
a) Subtract all multiples of p from 1 to n [all multiples of p
will have gcd more than 1 (at least p) with n]
b) Update n by repeatedly dividing it by p.
3) If the reduced n is more than 1, then remove all multiples
of n from result.

Below is the implementation of the above algorithm.

## C++

 // C++ program to calculate Euler's // Totient Function #include  using namespace std;   int phi(int n) {     // Initialize result as n     int result = n;         // Consider all prime factors of n      // and subtract their multiples      // from result     for(int p = 2; p * p <= n; ++p)     {                   // Check if p is a prime factor.         if (n % p == 0)          {                           // If yes, then update n and result             while (n % p == 0)                 n /= p;                               result -= result / p;         }     }        // If n has a prime factor greater than sqrt(n)     // (There can be at-most one such prime factor)     if (n > 1)         result -= result / n;               return result; }    // Driver code int main() {     int n;     for(n = 1; n <= 10; n++)     {         cout << "Phi" << "("              << n << ")" << " = "              << phi(n) << endl;     }     return 0; }   // This code is contributed by koulick_sadhu

## C

 // C program to calculate Euler's Totient Function #include    int phi(int n) {     int result = n; // Initialize result as n       // Consider all prime factors of n and subtract their     // multiples from result     for (int p = 2; p * p <= n; ++p) {                   // Check if p is a prime factor.         if (n % p == 0) {                           // If yes, then update n and result             while (n % p == 0)                 n /= p;             result -= result / p;         }     }       // If n has a prime factor greater than sqrt(n)     // (There can be at-most one such prime factor)     if (n > 1)         result -= result / n;     return result; }   // Driver program to test above function int main() {     int n;     for (n = 1; n <= 10; n++)         printf("phi(%d) = %d\n", n, phi(n));     return 0; }

## Java

 // Java program to calculate  // Euler's Totient Function import java.io.*;   class GFG  { static int phi(int n) {     // Initialize result as n     int result = n;        // Consider all prime factors      // of n and subtract their     // multiples from result     for (int p = 2; p * p <= n; ++p)     {                   // Check if p is          // a prime factor.         if (n % p == 0)          {                           // If yes, then update             // n and result             while (n % p == 0)                 n /= p;             result -= result / p;         }     }       // If n has a prime factor     // greater than sqrt(n)     // (There can be at-most      // one such prime factor)     if (n > 1)         result -= result / n;     return result; }   // Driver Code public static void main (String[] args) {     int n;     for (n = 1; n <= 10; n++)         System.out.println("phi(" + n +                             ") = " + phi(n)); } }   // This code is contributed by ajit

## Python3

 # Python3 program to calculate  # Euler's Totient Function def phi(n):           # Initialize result as n     result = n;        # Consider all prime factors     # of n and subtract their     # multiples from result     p = 2;      while(p * p <= n):                   # Check if p is a          # prime factor.         if (n % p == 0):                            # If yes, then              # update n and result             while (n % p == 0):                 n = int(n / p);             result -= int(result / p);         p += 1;       # If n has a prime factor     # greater than sqrt(n)     # (There can be at-most      # one such prime factor)     if (n > 1):         result -= int(result / n);     return result;   # Driver Code for n in range(1, 11):     print("phi(",n,") =", phi(n));       # This code is contributed  # by mits

## C#

 // C# program to calculate  // Euler's Totient Function using System;   class GFG {       static int phi(int n) { // Initialize result as n int result = n;    // Consider all prime   // factors of n and  // subtract their  // multiples from result for (int p = 2;          p * p <= n; ++p) {           // Check if p is      // a prime factor.     if (n % p == 0)      {                   // If yes, then update         // n and result         while (n % p == 0)             n /= p;         result -= result / p;     } }   // If n has a prime factor // greater than sqrt(n) // (There can be at-most  // one such prime factor) if (n > 1)     result -= result / n; return result; }   // Driver Code static public void Main () {     int n;     for (n = 1; n <= 10; n++)         Console.WriteLine("phi(" + n +                                ") = " +                               phi(n)); } }   // This code is contributed  // by akt_mit

## PHP

 <Φphp // PHP program to calculate  // Euler's Totient Function   function phi($n) {  // Initialize   // result as n  $result = $n;   // Consider all prime   // factors of n and subtract   // their multiples from result  for ($p = 2;           $p * $p <= $n; ++$p)     {                   // Check if p is          // a prime factor.         if ($n % $p == 0)          {                           // If yes, then              // update n and result             while ($n % $p == 0)                 $n = (int)$n / $p;  $result -= (int)$result / $p;         }     }       // If n has a prime factor     // greater than sqrt(n)     // (There can be at-most      // one such prime factor)     if ($n > 1)  $result -= (int)$result / $n;     return $result; } // Driver Code for ($n = 1; $n <= 10; $n++)     echo "phi(", $n,") =",   phi($n), "\n";       // This code is contributed  // by ajit Φ>

## Javascript

 // Javascript program to calculate  // Euler's Totient Function   function phi(n) {     // Initialize      // result as n     let result = n;        // Consider all prime      // factors of n and subtract      // their multiples from result     for (let p = 2;           p * p <= n; ++p)     {                   // Check if p is          // a prime factor.         if (n % p == 0)          {                           // If yes, then              // update n and result             while (n % p == 0)                 n = parseInt(n / p);             result -= parseInt(result / p);         }     }       // If n has a prime factor     // greater than sqrt(n)     // (There can be at-most      // one such prime factor)     if (n > 1)         result -= parseInt(result / n);     return result; }   // Driver Code for (let n = 1; n <= 10; n++)     document.write(phi(${n}) =${phi(n)}
);       // This code is contributed  // by _saurabh_jaiswal

Output

Phi(1) = 1
Phi(2) = 1
Phi(3) = 2
Phi(4) = 2
Phi(5) = 4
Phi(6) = 2
Phi(7) = 6
Phi(8) = 4
Phi(9) = 6
Phi(10) = 4

Time Complexity: O(√n log n)
Auxiliary Space: O(1)

Let us take an example to understand the above algorithm.

n = 10.
Initialize: result = 10

2 is a prime factor, so n = n/i = 5, result = 5
3 is not a prime factor.

The for loop stops after 3 as 4*4 is not less than or equal
to 10.

After for loop, result = 5, n = 5
Since n > 1, result = result - result/n = 4

## Some Interesting Properties of Euler’s Totient Function

1) For a prime number p Proof : , where p is any prime numberWe know that where k is any random number and [Tex]\\[/Tex]Total number from 1 to p = p Number for which is , i.e the number p itself, so subtracting 1 from p Examples : [Tex]\\[/Tex] [Tex]\\[/Tex] 2) For two prime numbers a and b , used in RSA Algorithm

Proof : , where a and b are prime numbers , [Tex]\\[/Tex]Total number from 1 to ab = ab Total multiples of a from 1 to ab = = Total multiples of b from 1 to ab = = Example:a = 5, b = 7, ab = 35Multiples of a = = 7 {5, 10, 15, 20, 25, 30, 35}Multiples of b = = 5 {7, 14, 21, 28, 35} Can there be any double counting ?(watch above example carefully, try with other prime numbers also for more grasp)Ofcourse, we have counted twice in multiples of a and multiples of b so, Total multiples =  a + b - 1 (with which with ) [Tex]\phi(ab) = ab - (a + b - 1)[/Tex] , removing all number with with  [Tex]\phi(ab) = (a - 1) \cdot (b - 1)[/Tex] Examples : [Tex]\\[/Tex] [Tex]\\[/Tex] 3) For a prime number p Proof : , where p is a prime number Total numbers from 1 to Total multiples of Removing these multiples as with them [Tex]\\[/Tex]Example : p = 2, k = 5, = 32Multiples of 2 (as with them ) = 32 / 2 = 16 {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32} [Tex]\phi(p ^ k) = p ^ k - p ^ {k - 1}[/Tex]

Examples : [Tex]\\[/Tex] [Tex]\\[/Tex] 4) For two number a and b   Special Case : gcd(a, b) = 1 Examples :

Special Case : ,  [Tex]\\[/Tex] [Tex]\\[/Tex]  [Tex]\\[/Tex]Normal Case : , [Tex]\\[/Tex]   [Tex]\\[/Tex] [Tex]\\[/Tex] 5) Sum of values of totient functions of all divisors of n is equal to n. Examples :

n = 6
factors = {1, 2, 3, 6}
n = = 1 + 1 + 2 + 2 = 6 n = 8factors = {1, 2, 4, 8}n = = 1 + 1 + 2 + 4 = 8 n = 10factors = {1, 2, 5, 10}n = = 1 + 1 + 4 + 4 = 10

6) The most famous and important feature is expressed in Euler’s theorem

The theorem states that if n and a are coprime
(or relatively prime) positive integers, then

aΦ(n) ≡ 1 (mod n) 

The RSA cryptosystem is based on this theorem:
In the particular case when m is prime say p, Euler’s theorem turns into the so-called Fermat’s little theorem

ap-1 ≡ 1 (mod p)

https://cp-algorithms.com/algebra/phi-function.html

http://mathcenter.oxford.memory.edu/site/math125/chineseRemainderTheorem/