Skip to content
Related Articles

Related Articles

Improve Article

Find the GCD of N Fibonacci Numbers with given Indices

  • Last Updated : 22 Apr, 2021

Given indices of N Fibonacci numbers. The task is to find the GCD of the Fibonacci numbers present at the given indices.
The first few Fibonacci numbers are: 
 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

Note: The indices start from zero. That is, 0th Fibonacci number = 0.
Examples: 
 

Input: Indices = {2, 3, 4, 5}
Output: GCD of the fibonacci numbers = 1

Input: Indices = {3, 6, 9} 
Output: GCD of the fibonacci numbers = 2

 

Brute force Approach: The brute force solution is to find all the Fibonacci numbers present at the given indices and compute the GCD of all of them, and print the result.
Efficient Approach: An efficient approach is to use the property: 
 



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

The idea is to calculate the GCD of all the indices and then find the Fibonacci number at the index gcd_1( where gcd_1 is the GCD of the given indices). 
Below is the implementation of the above approach: 
 

C++




// C++ program to Find the GCD of N Fibonacci
// Numbers with given Indices
#include <bits/stdc++.h>
using namespace std;
 
// Function to return n'th
// Fibonacci number
int getFib(int n)
{
    /* Declare an array to store Fibonacci numbers. */
    int f[n + 2]; // 1 extra to handle case, n = 0
    int i;
 
    // 0th and 1st number of the series
    // are 0 and 1
    f[0] = 0;
    f[1] = 1;
 
    for (i = 2; i <= n; i++) {
        // Add the previous 2 numbers in the series
        // and store it
        f[i] = f[i - 1] + f[i - 2];
    }
 
    return f[n];
}
 
// Function to Find the GCD of N Fibonacci
// Numbers with given Indices
int find(int arr[], int n)
{
    int gcd_1 = 0;
    // find the gcd of the indices
    for (int i = 0; i < n; i++) {
        gcd_1 = __gcd(gcd_1, arr[i]);
    }
 
    // find the fibonacci number at
    // index gcd_1
    return getFib(gcd_1);
}
 
// Driver code
int main()
{
    int indices[] = { 3, 6, 9 };
    int N = sizeof(indices) / sizeof(int);
 
    cout << find(indices, N);
 
    return 0;
}
Java



// Java program to Find the GCD of N Fibonacci
// Numbers with given Indices
import java.io.*;
 
// Function to return n'th
// Fibonacci number
 
public class GFG {
    // Recursive function to return gcd of a and b
    static int __gcd(int a, int b)
    {
        // Everything divides 0
        if (a == 0)
        return b;
        if (b == 0)
        return a;
         
        // base case
        if (a == b)
            return a;
         
        // a is greater
        if (a > b)
            return __gcd(a-b, b);
        return __gcd(a, b-a);
    }
 
static int getFib(int n)
{
    /* Declare an array to store Fibonacci numbers. */
    int f[] = new int[n + 2];
    // 1 extra to handle case, n = 0
    int i;
 
    // 0th and 1st number of the series
    // are 0 and 1
    f[0] = 0;
    f[1] = 1;
 
    for (i = 2; i <= n; i++) {
        // Add the previous 2 numbers in the series
        // and store it
        f[i] = f[i - 1] + f[i - 2];
    }
 
    return f[n];
}
 
// Function to Find the GCD of N Fibonacci
// Numbers with given Indices
static int find(int arr[], int n)
{
    int gcd_1 = 0;
    // find the gcd of the indices
    for (int i = 0; i < n; i++) {
        gcd_1 = __gcd(gcd_1, arr[i]);
    }
 
    // find the fibonacci number at
    // index gcd_1
    return getFib(gcd_1);
}
 
// Driver code
    public static void main (String[] args) {
        int indices[] = { 3, 6, 9 };
    int N = indices.length;
 
    System.out.println( find(indices, N));
    }
}
Python 3



# Python program to Find the
# GCD of N Fibonacci Numbers
# with given Indices
from math import *
 
# Function to return n'th
# Fibonacci number
def getFib(n) :
 
    # Declare an array to store
    # Fibonacci numbers.
    f = [0] * (n + 2) # 1 extra to handle case, n = 0
 
    # 0th and 1st number of the
    # series are 0 and 1
    f[0], f[1] = 0, 1
 
    # Add the previous 2 numbers
    # in the series and store it
    for i in range(2, n + 1) :
 
        f[i] = f[i - 1] + f[i - 2]
 
    return f[n]
 
# Function to Find the GCD of N Fibonacci
# Numbers with given Indices
def find(arr, n) :
 
    gcd_1 = 0
 
    # find the gcd of the indices
    for i in range(n) :
        gcd_1 = gcd(gcd_1, arr[i])
 
    # find the fibonacci number
    # at index gcd_1
    return getFib(gcd_1)
 
# Driver code    
if __name__ == "__main__" :
 
    indices = [3, 6, 9]
    N = len(indices)
 
    print(find(indices, N))
 
# This code is contributed by ANKITRAI1
C#



// C# program to Find the GCD
// of N Fibonacci Numbers with
// given Indices
using System;
 
// Function to return n'th
// Fibonacci number
class GFG
{
// Recursive function to
// return gcd of a and b
static int __gcd(int a, int b)
{
    // Everything divides 0
    if (a == 0)
    return b;
    if (b == 0)
    return a;
     
    // base case
    if (a == b)
        return a;
     
    // a is greater
    if (a > b)
        return __gcd(a - b, b);
    return __gcd(a, b - a);
}
 
static int getFib(int n)
{
    /* Declare an array to
    store Fibonacci numbers. */
    int []f = new int[n + 2];
     
    // 1 extra to handle case, n = 0
    int i;
 
    // 0th and 1st number of
    // the series are 0 and 1
    f[0] = 0;
    f[1] = 1;
 
    for (i = 2; i <= n; i++)
    {
        // Add the previous 2 numbers
        // in the series and store it
        f[i] = f[i - 1] + f[i - 2];
    }
 
    return f[n];
}
 
// Function to Find the GCD
// of N Fibonacci Numbers
// with given Indices
static int find(int []arr, int n)
{
    int gcd_1 = 0;
     
    // find the gcd of the indices
    for (int i = 0; i < n; i++)
    {
        gcd_1 = __gcd(gcd_1, arr[i]);
    }
 
    // find the fibonacci number
    // at index gcd_1
    return getFib(gcd_1);
}
 
// Driver code
public static void Main ()
{
    int []indices = { 3, 6, 9 };
    int N = indices.Length;
 
    Console.WriteLine(find(indices, N));
}
}
 
// This code is contributed
// by Shashank
PHP



<?php
// PHP program to Find the GCD of
// N Fibonacci Numbers with given
// Indices
 
// Function to return n'th
// Fibonacci number
function gcd($a, $b)
{
    return $b ? gcd($b, $a % $b) : $a;
}
 
function getFib($n)
{
    /* Declare an array to store
    Fibonacci numbers. */
     
    // 1 extra to handle case, n = 0
    $f = array_fill(0, ($n + 2), NULL);
 
    // 0th and 1st number of the
    // series are 0 and 1
    $f[0] = 0;
    $f[1] = 1;
 
    for ($i = 2; $i <= $n; $i++)
    {
        // Add the previous 2 numbers
        // in the series and store it
        $f[$i] = $f[$i - 1] + $f[$i - 2];
    }
 
    return $f[$n];
}
 
// Function to Find the GCD of N Fibonacci
// Numbers with given Indices
function find(&$arr, $n)
{
    $gcd_1 = 0;
     
    // find the gcd of the indices
    for ($i = 0; $i < $n; $i++)
    {
        $gcd_1 = gcd($gcd_1, $arr[$i]);
    }
 
    // find the fibonacci number
    // at index gcd_1
    return getFib($gcd_1);
}
 
// Driver code
$indices = array(3, 6, 9 );
$N = sizeof($indices);
 
echo find($indices, $N);
 
// This code is contributed
// by ChitraNayal
?>
Javascript



<script>
 
// javascript program to
// Find the GCD of N Fibonacci
// Numbers with given Indices
 
// Function to return n'th
// Fibonacci number
 
    // Recursive function to return gcd of a and b
    function __gcd(a , b) {
        // Everything divides 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;
 
        // base case
        if (a == b)
            return a;
 
        // a is greater
        if (a > b)
            return __gcd(a - b, b);
        return __gcd(a, b - a);
    }
 
    function getFib(n) {
        /* Declare an array to store Fibonacci numbers. */
        var f = Array(n + 2).fill(0);
        // 1 extra to handle case, n = 0
        var i;
 
        // 0th and 1st number of the series
        // are 0 and 1
        f[0] = 0;
        f[1] = 1;
 
        for (i = 2; i <= n; i++) {
            // Add the previous 2 numbers in the series
            // and store it
            f[i] = f[i - 1] + f[i - 2];
        }
 
        return f[n];
    }
 
    // Function to Find the GCD of N Fibonacci
    // Numbers with given Indices
    function find(arr , n) {
        var gcd_1 = 0;
        // find the gcd of the indices
        for (i = 0; i < n; i++) {
            gcd_1 = __gcd(gcd_1, arr[i]);
        }
 
        // find the fibonacci number at
        // index gcd_1
        return getFib(gcd_1);
    }
 
    // Driver code
     
        var indices = [ 3, 6, 9 ];
        var N = indices.length;
 
        document.write(find(indices, N));
 
// This code contributed by gauravrajput1
 
</script>
Output: 
2
 
Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.



My Personal Notes
arrow_drop_up
Previous
Find nth Term of the Series 1 2 2 4 4 4 4 8 8 8 8 8 8 8 8 ...
Next

GCD and Fibonacci Numbers
Recommended Articles
Page :
Count pairs of indices having sum of indices same as the sum of elements at those indices
21, Jul 21
Count of indices pairs such that product of elements at these indices is equal to absolute difference of indices
07, Jul 21
Count of unordered pair of indices such that ratio of elements at these indices is same as ratio of indices
24, Aug 21
Count of numbers of length N having prime numbers at odd indices and odd numbers at even indices
25, Aug 21
GCD and Fibonacci Numbers
19, Oct 16
Check if GCD of all Composite Numbers in an array divisible by K is a Fibonacci Number or not
24, Nov 20
Find the Array Permutation having sum of elements at odd indices greater than sum of elements at even indices
07, Jul 21
Find a permutation such that number of indices for which gcd(p[i], i) > 1 is exactly K
12, Feb 19
Smallest subsequence having GCD equal to GCD of given array
20, Oct 20
Pair of integers having least GCD among all given pairs having GCD exceeding K
16, Sep 20
Longest increasing subsequence consisting of elements from indices divisible by previously selected indices
04, Dec 20
Count pairs with product of indices equal to the product of elements present at those indices
18, Mar 21
Rearrange sorted array such that all odd indices elements comes before all even indices element
09, Sep 21
Count of triplets whose indices and element at that indices are in increasing order
15, Sep 21
Check if a M-th fibonacci number divides N-th fibonacci number
11, Jul 18
Check if sum of Fibonacci elements in an Array is a Fibonacci number or not
09, Apr 20
Rearrange array to maximize sum of GCD of array elements with their respective indices
04, Jan 21
GCD of elements occurring Fibonacci number of times in an Array
09, Apr 20
Count all possible values of K less than Y such that GCD(X, Y) = GCD(X+K, Y)
13, Jan 21
Find two numbers whose sum and GCD are given
21, Nov 18
Print N lines of 4 numbers such that every pair among 4 numbers has a GCD K
27, Oct 18
Find the numbers present at Kth level of a Fibonacci Binary Tree
07, Apr 20
Find two Fibonacci numbers whose sum can be represented as N
08, Apr 20
Find the sum of first N odd Fibonacci numbers
15, Nov 18
Article Contributed By :
https://media.geeksforgeeks.org/auth/avatar.png
andrew1234
@andrew1234
Vote for difficulty





Improved By :
Article Tags :
Practice Tags :
Report Issue

We use cookies to ensure you have the best browsing experience on our website. By using our site, you
acknowledge that you have read and understood our
Cookie Policy &
        Privacy Policy

Lightbox