Given two integers n and k. Find position the n’th multiple of K in the Fibonacci series.

**Examples:**

Input : k = 2, n = 3 Output : 9 3'rd multiple of 2 in Fibonacci Series is 34 which appears at position 9. Input : k = 4, n = 5 Output : 30 5'th multiple of 4 in Fibonacci Series is 832040 which appears at position 30.

An **Efficient Solution **is based on below interesting property.

Fibonacci series is always periodic under modular representation. Below are examples.

F (mod 2) = 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0 Here 0 is repeating at every 3rd index and the cycle repeats at every 3rd index. F (mod 3) = 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2 Here 0 is repeating at every 4th index and the cycle repeats at every 8th index. F (mod 4) = 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0 Here 0 is repeating at every 6th index and the cycle repeats at every 6th index. F (mod 5) = 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0, 1, 1, 2, 3, 0, 3, 3, 1, 4, 0 Here 0 is repeating at every 5th index and the cycle repeats at every 20th index. F (mod 6) = 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 0, 1, 1, 2, 3, 5, 2 Here 0 is repeating at every 12th index and the cycle repeats at every 24th index. F (mod 7) = 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6 Here 0 is repeating at every 8th index and the cycle repeats at every 16th index. F (mod 8) = 1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1, 0, 1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1, 0, 1, 1, 2, 3, 5, 0 Here 0 is repeating at every 6th index and the cycle repeats at every 12th index. F (mod 9) = 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 0, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 0, 1, 1, 2, 3, 5, 8 Here 0 is repeating at every 12th index and the cycle repeats at every 24th index. F (mod 10) = 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0. Here 0 is repeating at every 15th index and the cycle repeats at every 60th index.

## C++

`// C++ program to find position of n'th multiple` `// of a mumber k in Fibonacci Series` `#include <bits/stdc++.h>` `using` `namespace` `std;` `const` `int` `MAX = 1000;` `// Returns position of n'th multple of k in` `// Fibonacci Series` `int` `findPosition(` `int` `k, ` `int` `n)` `{` ` ` `// Iterate through all fibonacci numbers` ` ` `unsigned ` `long` `long` `int` `f1 = 0, f2 = 1, f3;` ` ` `for` `(` `int` `i = 2; i <= MAX; i++) {` ` ` `f3 = f1 + f2;` ` ` `f1 = f2;` ` ` `f2 = f3;` ` ` `// Found first multiple of k at position i` ` ` `if` `(f2 % k == 0)` ` ` `// n'th multiple would be at position n*i` ` ` `// using Periodic property of Fibonacci` ` ` `// numbers under modulo.` ` ` `return` `n * i;` ` ` `}` `}` `// Driver Code` `int` `main()` `{` ` ` `int` `n = 5, k = 4;` ` ` `cout << ` `"Position of n'th multiple of k"` ` ` `<< ` `" in Fibonacci Series is "` ` ` `<< findPosition(k, n) << endl;` ` ` `return` `0;` `}` |

*chevron_right*

*filter_none*

**Output:**

Position of n'th multiple of k in Fibonacci Series is 30

Please refer complete article on n’th multiple of a number in Fibonacci Series for more details!

## Recommended Posts:

- n'th multiple of a number in Fibonacci Series
- Sum of nth terms of Modified Fibonacci series made by every pair of two arrays
- Find the Nth element of the modified Fibonacci series
- Nth Term of a Fibonacci Series of Primes formed by concatenating pairs of Primes in a given range
- Program to find last two digits of Nth Fibonacci number
- Program to find Nth odd Fibonacci Number
- Check if a M-th fibonacci number divides N-th fibonacci number
- Check if sum of Fibonacci elements in an Array is a Fibonacci number or not
- G-Fact 18 | Finding nth Fibonacci Number using Golden Ratio
- Nth Even Fibonacci Number
- Find nth Fibonacci number using Golden ratio
- Nth XOR Fibonacci number
- Nth Fibonacci number using Pell's equation
- Fast Doubling method to find the Nth Fibonacci number
- An efficient way to check whether n-th Fibonacci number is multiple of 10
- C++ Program to print Fibonacci Series using Class template
- C Program to print Fibonacci Series without using loop
- Program to find Nth term in the series 0, 2, 1, 3, 1, 5, 2, 7, 3,…
- Factorial of each element in Fibonacci series
- Print Fibonacci Series in reverse order