💬 ชาวเน็ตแห่แชร์: web//https://www.geeksforgeeks.org/fermats little theorem

Fermat’s little theorem states that if p is a prime number, then for any integer a, the number a ^p – a is an integer multiple of p.

Here p is a prime number
a^p ≡ a (mod p).

Special Case: If a is not divisible by p, Fermat’s little theorem is equivalent to the statement that a ^p-1-1 is an integer multiple of p.

a^p-1 ≡ 1 (mod p)
OR
a^p-1 % p = 1
Here a is not divisible by p.

Take an Example How Fermat’s little theorem works
Examples:

 P = an integer Prime number   
 a = an integer which is not multiple of P  
 Let a = 2 and P = 17 
 
 According to Fermat's little theorem 
  2^{17 - 1}    ≡ 1 mod(17)
 we got  65536 % 17 ≡ 1   
 that mean (65536-1) is an multiple of 17

Use of Fermat’s little theorem

If we know m is prime, then we can also use Fermats’s little theorem to find the inverse.

a^m-1 ≡ 1 (mod m)
If we multiply both sides with a^-1, we get

a^-1 ≡ a ^m-2 (mod m)
Below is the Implementation of above

C++

// C++ program to find modular inverse of a 
// under modulo m using Fermat's little theorem. 
// This program works only if m is prime. 
#include <bits/stdc++.h> 
using namespace std; 
  
// To compute x raised to power y under modulo m 
int power(int x, unsigned int y, unsigned int m); 
  
// Function to find modular inverse of a under modulo m 
// Assumption: m is prime 
void modInverse(int a, int m) 
{ 
    if (__gcd(a, m) != 1) 
        cout << "Inverse doesn't exist"; 
  
    else { 
  
        // If a and m are relatively prime, then 
        // modulo inverse is a^(m-2) mode m 
        cout << "Modular multiplicative inverse is "
             << power(a, m - 2, m); 
    } 
} 
  
// To compute x^y under modulo m 
int power(int x, unsigned int y, unsigned int m) 
{ 
    if (y == 0) 
        return 1; 
    int p = power(x, y / 2, m) % m; 
    p = (p * p) % m; 
  
    return (y % 2 == 0) ? p : (x * p) % m; 
} 
  
// Driver Program 
int main() 
{ 
    int a = 3, m = 11; 
    modInverse(a, m); 
    return 0; 
} 

Java

// Java program to find modular  
// inverse of a under modulo m  
// using Fermat's little theorem.  
// This program works only if m is prime. 
  
class GFG 
{ 
    static int __gcd(int a, int b) 
    { 
      
        if(b == 0)  
        { 
            return a; 
        } 
        else 
        { 
            return __gcd(b, a % b); 
        } 
    } 
      
    // To compute x^y under modulo m 
    static int power(int x,int y,int m) 
    { 
        if (y == 0) 
            return 1; 
        int p = power(x, y / 2, m) % m; 
        p = (p * p) % m; 
      
        return (y % 2 == 0) ? p : (x * p) % m; 
    } 
      
    // Function to find modular  
    // inverse of a under modulo m 
    // Assumption: m is prime 
    static void modInverse(int a, int m) 
    { 
        if (__gcd(a, m) != 1) 
            System.out.print("Inverse doesn't exist"); 
      
        else { 
      
            // If a and m are relatively prime, then 
            // modulo inverse is a^(m-2) mode m 
            System.out.print("Modular multiplicative inverse is "
                                            +power(a, m - 2, m)); 
        } 
    } 
      
      
    // Driver code 
    public static void main (String[] args)  
    { 
        int a = 3, m = 11; 
        modInverse(a, m); 
    } 
} 
  
// This code is contributed by Anant Agarwal. 

Python3

# Python program to find 
# modular inverse of a 
# under modulo m using 
# Fermat's little theorem. 
# This program works 
# only if m is prime. 
  
def __gcd(a,b): 
  
    if(b == 0): 
        return a 
    else: 
        return __gcd(b, a % b) 
      
# To compute x^y under modulo m 
def power(x,y,m): 
  
    if (y == 0): 
        return 1
    p = power(x, y // 2, m) % m 
    p = (p * p) % m 
   
    return p if(y % 2 == 0) else  (x * p) % m 
  
# Function to find modular 
# inverse of a under modulo m 
# Assumption: m is prime 
def modInverse(a,m): 
  
    if (__gcd(a, m) != 1): 
        print("Inverse doesn't exist") 
   
    else: 
   
        # If a and m are relatively prime, then 
        # modulo inverse is a^(m-2) mode m 
        print("Modular multiplicative inverse is ", 
             power(a, m - 2, m)) 
  
# Driver code 
  
a = 3
m = 11
modInverse(a, m) 
  
# This code is contributed 
# by Anant Agarwal. 

C#

// C# program to find modular  
// inverse of a under modulo m  
// using Fermat's little theorem.  
// This program works only if m is prime. 
using System; 
  
class GFG 
{ 
    static int __gcd(int a, int b) 
    { 
      
        if(b == 0)  
        { 
            return a; 
        } 
        else
        { 
            return __gcd(b, a % b); 
        } 
    } 
      
    // To compute x^y under modulo m 
    static int power(int x, int y, int m) 
    { 
        if (y == 0) 
            return 1; 
        int p = power(x, y / 2, m) % m; 
        p = (p * p) % m; 
      
        return (y % 2 == 0) ? p : (x * p) % m; 
    } 
      
    // Function to find modular  
    // inverse of a under modulo m 
    // Assumption: m is prime 
    static void modInverse(int a, int m) 
    { 
        if (__gcd(a, m) != 1) 
            Console.WriteLine("Modular multiplicative inverse is "
                                            +power(a, m - 2, m)); 
      
        else { 
      
            // If a and m are relatively prime, then 
            // modulo inverse is a^(m-2) mode m 
            Console.WriteLine("Modular multiplicative inverse is "
                                            +power(a, m - 2, m)); 
        } 
    } 
      
      
    // Driver code 
    public static void Main ()  
    { 
        int a = 3, m = 11; 
        modInverse(a, m); 
    } 
} 
  
// This code is contributed by vt_m. 

PHP

<?php 
// PHP program to find modular inverse of a 
// under modulo m using Fermat's little theorem. 
// This program works only if m is prime. 
  
  
// To compute x raised to 
// power y under modulo m 
// Recursive function to  
// return gcd of a and b 
function __gcd($a, $b) 
{ 
      
    // Everything divides 0  
    if ($a == 0 || $b == 0) 
    return 0; 
  
    // base case 
    if ($a == $b) 
        return $a; 
  
    // a is greater 
    if ($a > $b) 
        return __gcd($a-$b, $b); 
    return __gcd($a, $b-$a); 
} 
  
// Function to find modular 
// inverse of a under modulo m 
// Assumption: m is prime 
function modInverse($a, $m) 
{ 
    if (__gcd($a, $m) != 1) 
        echo "Inverse doesn't exist"; 
  
    else 
    { 
  
        // If a and m are relatively 
        // prime, then modulo inverse 
        // is a^(m-2) mode m 
        echo "Modular multiplicative inverse is ", 
                             power($a,$m - 2, $m); 
    } 
} 
  
// To compute x^y under modulo m 
function power($x, $y, $m) 
{ 
    if ($y == 0) 
        return 1; 
    $p = power($x,$y / 2, $m) % $m; 
    $p = ($p * $p) % $m; 
  
    return ($y % 2 == 0) ? $p : ($x * $p) % $m; 
} 
  
    // Driver Code 
    $a = 3; $m = 11; 
    modInverse($a, $m); 
      
// This code is contributed by anuj__67. 
?> 

Output :

Modular multiplicative inverse is 4

Some Article Based on Fermat’s little theorem

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