# Fermat’s 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
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. ` `?> `

Output :

```Modular multiplicative inverse is 4
```

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