Find the GCD of N Fibonacci Numbers with given Indices

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

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 
?> 

Output:

My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python 3

C#

PHP

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