Given two integers n and k. Find position the n’th multiple of K in the Fibonacci series.
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 5 in Fibonacci Series is 832040 which appears at position 30.
Fibonacci Series(F) : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040… (neglecting the first 0).
A Simple Solution is to traverse Fibonacci numbers starting from first number. While traversing, keep track of counts of multiples of k. Whenever the count becomes n, return the position.
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.
Why is Fibonacci Series Periodic under Modulo?
Under modular representation, we know that each Fibonacci number will be represented as some residue 0 ? F (mod m) < m. Thus, there are only m possible values for any given F (mod m) and hence m*m = m^2 possible pairs of consecutive terms within the sequence. Since m^2 is finite, we know that some pair of terms must eventually repeat itself. Also, as any pair of terms in the Fibonacci sequence determines the rest of the sequence, we see that the Fibonacci series modulo m must repeat itself at some point, and thus must be periodic.
Source : https://www.whitman.edu/Documents/Academics/Mathematics/clancy.pdf
Based on above fact, we can quickly find position of n’th multiple of K by simply finding first multiple. If position of first multiple is i, we return position as n*i.
Below is the implementation :
Position of n'th multiple of k in Fibonacci Series is 30
This article is contributed by Kishlay Verma. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
- C/C++ Program for nth multiple of a number in Fibonacci Series
- An efficient way to check whether n-th Fibonacci number is multiple of 10
- K- Fibonacci series
- Check if a M-th fibonacci number divides N-th fibonacci number
- Factorial of each element in Fibonacci series
- Print Fibonacci Series in reverse order
- Sum of nth terms of Modified Fibonacci series made by every pair of two arrays
- Find n terms of Fibonacci type series with given first two terms
- Number of ways to represent a number as sum of k fibonacci numbers
- Finding number of digits in n'th Fibonacci number
- Nth Even Fibonacci Number
- Check if a number is multiple of 5 without using / and % operators
- Round the given number to nearest multiple of 10
- Fibonacci number in an array
- Check whether a very large number of the given form is a multiple of 3.