Python Program for n\’th multiple of a number in Fibonacci Series
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 5 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.
Python3
# Python Program to find position of n\'th multiple # of a number k in Fibonacci Series def findPosition(k, n): f1 = 0 f2 = 1 i = 2 ; while i! = 0 : f3 = f1 + f2; f1 = f2; f2 = f3; if f2 % k = = 0 : return n * i i + = 1 return # Multiple no. n = 5 ; # Number of whose multiple we are finding k = 4 ; print ( "Position of n\'th multiple of k in" "Fibonacci Series is" , findPosition(k,n)); # Code contributed by Mohit Gupta_OMG |
Output:
Position of n'th multiple of k inFibonacci Series is 30
Please refer complete article on n’th multiple of a number in Fibonacci Series for more details!