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 ?> |
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