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Fermat’s little theorem
• Difficulty Level : Medium
• Last Updated : 20 Apr, 2021

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
ap ≡ 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.

ap-1 ≡ 1 (mod p)
OR
ap-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.
am-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 ``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

 ` ``\$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.``?>`

## Javascript

 ``

Output :

`Modular multiplicative inverse is 4`

Some Article Based on Fermat’s little theorem

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