Java Program for nth 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.
// Java Program to find position of n'th multiple // of a mumber k in Fibonacci Series package gfg; public class GFG { public static int findPosition(int k, int n) { long f1 = 0, f2 = 1, f3; int i = 2; while (i != 0) { f3 = f1 + f2; f1 = f2; f2 = f3; if (f2 % k == 0) { return n * i; } i++; } return 0; } public static void main(String[] args) { // Multiple no. int n = 5; // Number of whose multiple we are finding int k = 4; System.out.print("Position of n'th multiple" + " of k in Fibonacci Series is "); System.out.println(findPosition(k, n)); } } // Code contributed by Mohit Gupta_OMG |
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Output:
Please refer complete article on n’th multiple of a number in Fibonacci Series for more details!
