# 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
```

## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

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://www.geeksforgeeks.org/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://www.geeksforgeeks.org/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://www.geeksforgeeks.org/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://www.geeksforgeeks.org/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
```

This article is contributed by Shubham Agrawal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.

My Personal Notes arrow_drop_up

Improved By : nitin mittal, Mithun Kumar

Article Tags :
Practice Tags :

3

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.