# GCD and Fibonacci Numbers

You are given two positive numbers M and N. The task is to print greatest common divisor of M’th and N’th Fibonacci Numbers.

The first few Fibonacci Numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ….
Note that 0 is considered as 0’th Fibonacci Number.

Examples:

```Input  : M = 3, N = 6
Output :  2
Fib(3) = 2, Fib(6) = 8
GCD of above two numbers is 2

Input  : M = 8, N = 12
Output :  3
Fib(8) = 21, Fib(12) = 144
GCD of above two numbers is 3
```

A Simple Solution is to follow below steps.
1) Find M’th Fibonacci Number.
2) Find N’th Fibonacci Number.
3) Return GCD of two numbers.

A Better Solution is based on below identity

```GCD(Fib(M), Fib(N)) = Fib(GCD(M, N))

The above property holds because Fibonacci Numbers follow
Divisibility Sequence, i.e., if M divides N, then Fib(M)
also divides N. For example, Fib(3) = 2 and every third
third Fibonacci Number is even.

Source : Wiki
```

The steps are:
1) Find GCD of M and N. Let GCD be g.
2) Return Fib(g).

Below are implementations of above idea.

## C++

 `// C++ Program to find GCD of Fib(M) and Fib(N) ` `#include ` `using` `namespace` `std; ` `const` `int` `MAX = 1000; ` ` `  `// Create an array for memoization ` `int` `f[MAX] = {0}; ` ` `  `// Returns n'th Fibonacci number using table f[].  ` `// Refer method 6 of below post for details. ` `// https://tutorialspoint.dev/slugresolver/program-for-nth-fibonacci-number/ ` `int` `fib(``int` `n) ` `{ ` `    ``// Base cases ` `    ``if` `(n == 0) ` `        ``return` `0; ` `    ``if` `(n == 1 || n == 2) ` `        ``return` `(f[n] = 1); ` ` `  `    ``// If fib(n) is already computed ` `    ``if` `(f[n]) ` `        ``return` `f[n]; ` ` `  `    ``int` `k = (n & 1)? (n+1)/2 : n/2; ` ` `  `    ``// Applying recursive formula [Note value n&1 is 1 ` `    ``// if n is odd, else 0. ` `    ``f[n] = (n & 1)? (fib(k)*fib(k) + fib(k-1)*fib(k-1)) ` `           ``: (2*fib(k-1) + fib(k))*fib(k); ` ` `  `    ``return` `f[n]; ` `} ` ` `  `// Function to return gcd of a and b ` `int` `gcd(``int` `M, ``int` `N) ` `{ ` `    ``if` `(M == 0) ` `        ``return` `N; ` `    ``return` `gcd(N%M, M); ` `} ` ` `  `// Returns GCD of Fib(M) and Fib(N) ` `int` `findGCDofFibMFibN(``int` `M,  ``int` `N) ` `{ ` `    ``return` `fib(gcd(M, N)); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `   ``int` `M = 3, N = 12; ` `   ``cout << findGCDofFibMFibN(M, N); ` `   ``return` `0; ` `} `

## Java

 `// Java Program to find GCD of Fib(M) and Fib(N) ` `class` `gcdOfFibonacci ` `{ ` `    ``static` `final` `int` `MAX = ``1000``; ` `    ``static` `int``[] f; ` ` `  `    ``gcdOfFibonacci()  ``// Constructor ` `    ``{ ` `        ``// Create an array for memoization ` `        ``f = ``new` `int``[MAX]; ` `    ``} ` ` `  `    ``// Returns n'th Fibonacci number using table f[]. ` `    ``// Refer method 6 of below post for details. ` `    ``// https://tutorialspoint.dev/slugresolver/program-for-nth-fibonacci-number/ ` `    ``private` `static` `int` `fib(``int` `n) ` `    ``{ ` `        ``// Base cases ` `        ``if` `(n == ``0``) ` `            ``return` `0``; ` `        ``if` `(n == ``1` `|| n == ``2``) ` `            ``return` `(f[n] = ``1``); ` ` `  `        ``// If fib(n) is already computed ` `        ``if` `(f[n]!=``0``) ` `            ``return` `f[n]; ` ` `  `        ``int` `k = ((n & ``1``)==``1``)? (n+``1``)/``2` `: n/``2``; ` ` `  `        ``// Applying recursive formula [Note value n&1 is 1 ` `        ``// if n is odd, else 0. ` `        ``f[n] = ((n & ``1``)==``1``)? (fib(k)*fib(k) + fib(k-``1``)*fib(k-``1``)) ` `               ``: (``2``*fib(k-``1``) + fib(k))*fib(k); ` ` `  `        ``return` `f[n]; ` `    ``} ` ` `  `    ``// Function to return gcd of a and b ` `    ``private` `static` `int` `gcd(``int` `M, ``int` `N) ` `    ``{ ` `        ``if` `(M == ``0``) ` `            ``return` `N; ` `        ``return` `gcd(N%M, M); ` `    ``} ` ` `  `    ``// This method returns GCD of Fib(M) and Fib(N) ` `    ``static` `int` `findGCDofFibMFibN(``int` `M,  ``int` `N) ` `    ``{ ` `        ``return` `fib(gcd(M, N)); ` `    ``} ` ` `  `    ``// Driver method ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``// Returns GCD of Fib(M) and Fib(N) ` `        ``gcdOfFibonacci obj = ``new` `gcdOfFibonacci(); ` `        ``int` `M = ``3``, N = ``12``; ` `        ``System.out.println(findGCDofFibMFibN(M, N)); ` `    ``} ` `} ` `// This code is contributed by Pankaj Kumar `

## Python3

 `# Python Program to find ` `# GCD of Fib(M) and Fib(N) ` ` `  `MAX` `=` `1000` `  `  `# Create an array for memoization ` `f``=``[``0` `for` `i ``in` `range``(``MAX``)] ` `  `  `# Returns n'th Fibonacci ` `# number using table f[].  ` `# Refer method 6 of below ` `# post for details. ` `# https://tutorialspoint.dev/slugresolver/program-for-nth-fibonacci-number/ ` `def` `fib(n): ` ` `  `    ``# Base cases ` `    ``if` `(n ``=``=` `0``): ` `        ``return` `0` `    ``if` `(n ``=``=` `1` `or` `n ``=``=` `2``): ` `        ``f[n] ``=` `1` `  `  `    ``# If fib(n) is already computed ` `    ``if` `(f[n]): ` `        ``return` `f[n] ` `  `  `    ``k ``=` `(n``+``1``)``/``/``2` `if``(n & ``1``) ``else` `n``/``/``2` `  `  `    ``# Applying recursive ` `    ``# formula [Note value n&1 is 1 ` `    ``# if n is odd, else 0. ` `    ``f[n] ``=` `(fib(k)``*``fib(k) ``+` `fib(k``-``1``)``*``fib(k``-``1``)) ``if``(n & ``1``) ``else` `((``2``*` `           ``fib(k``-``1``) ``+` `fib(k))``*``fib(k)) ` `  `  `    ``return` `f[n] ` ` `  `  `  `# Function to return ` `# gcd of a and b ` `def` `gcd(M, N): ` ` `  `    ``if` `(M ``=``=` `0``): ` `        ``return` `N ` `    ``return` `gcd(N ``%` `M, M) ` ` `  `  `  `# Returns GCD of ` `# Fib(M) and Fib(N) ` `def` `findGCDofFibMFibN(M, N): ` ` `  `    ``return` `fib(gcd(M, N)) ` ` `  `  `  `# Driver code ` ` `  `M ``=` `3` `N ``=` `12` ` `  `print``(findGCDofFibMFibN(M, N)) ` ` `  `# This code is contributed ` `# by Anant Agarwal. `

## C#

 `// C# Program to find GCD of  ` `// Fib(M) and Fib(N) ` `using` `System; ` ` `  `class` `gcdOfFibonacci { ` `     `  `    ``static` `int` `MAX = 1000; ` `    ``static` `int` `[]f; ` ` `  `    ``// Constructor ` `    ``gcdOfFibonacci()  ` `    ``{ ` `        ``// Create an array ` `        ``// for memoization ` `        ``f = ``new` `int``[MAX]; ` `    ``} ` ` `  `    ``// Returns n'th Fibonacci number ` `    ``// using table f[]. Refer method  ` `    ``// 6 of below post for details. ` `    ``// https://tutorialspoint.dev/slugresolver/program-for-nth-fibonacci-number/ ` `    ``private` `static` `int` `fib(``int` `n) ` `    ``{ ` `        ``// Base cases ` `        ``if` `(n == 0) ` `            ``return` `0; ` `        ``if` `(n == 1 || n == 2) ` `            ``return` `(f[n] = 1); ` ` `  `        ``// If fib(n) is  ` `        ``// already computed ` `        ``if` `(f[n]!=0) ` `            ``return` `f[n]; ` ` `  `        ``int` `k = ((n & 1)==1)? (n+1)/2 : n/2; ` ` `  `        ``// Applying recursive formula  ` `        ``// [Note value n&1 is 1 ` `        ``// if n is odd, else 0. ` `        ``f[n] = ((n & 1) == 1) ? (fib(k) * fib(k) + ` `               ``fib(k - 1) * fib(k - 1)) :  ` `               ``(2 * fib(k - 1) + fib(k)) * fib(k); ` ` `  `        ``return` `f[n]; ` `    ``} ` ` `  `    ``// Function to return gcd of a and b ` `    ``private` `static` `int` `gcd(``int` `M, ``int` `N) ` `    ``{ ` `        ``if` `(M == 0) ` `            ``return` `N; ` `        ``return` `gcd(N%M, M); ` `    ``} ` ` `  `    ``// This method returns GCD of ` `    ``// Fib(M) and Fib(N) ` `    ``static` `int` `findGCDofFibMFibN(``int` `M, ``int` `N) ` `    ``{ ` `        ``return` `fib(gcd(M, N)); ` `    ``} ` ` `  `    ``// Driver method ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``// Returns GCD of Fib(M) and Fib(N) ` `        ``new` `gcdOfFibonacci(); ` `        ``int` `M = 3, N = 12; ` `        ``Console.Write(findGCDofFibMFibN(M, N)); ` `    ``} ` `} ` ` `  `// This code is contributed by nitin mittal. `

## PHP

 ` `

Output:

```2
```