查找具有给定索引的N个斐波纳契数的GCD

给定N个斐波纳契数的索引。任务是找到给定索引处的斐波那契数的GCD。
前几个斐波那契数是：

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

注意：索引从零开始。也就是说，第0个斐波那契数= 0。
例子：

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

蛮力方法：蛮力解决方案是找到给定索引上存在的所有斐波那契数，并计算所有这些数的GCD，然后打印结果。
高效的方法：高效的方法是使用该属性：

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

想法是计算所有索引的GCD，然后在索引gcd_1(其中gcd_1是给定索引的GCD)处找到斐波那契数。
下面是上述方法的实现：

C++

// C++ program to Find the GCD of N Fibonacci
// Numbers with given Indices
#include 
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

Javascript

 
// 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
 

输出：

2