# Python Program for nth multiple of a number in Fibonacci Series

Given two integers n and k. Find position the nth multiple of K in the Fibonacci series. **Examples:**

Input:k = 2, n = 3Output:9, 3rd multiple of 2 in Fibonacci Series is 34 that appears at position 9.

Input:k = 4, n = 5Output:30, 5th multiple of 5 in Fibonacci Series is 832040 which appears at position 30.

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

The 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.

Below is the implementation of above approach:

## 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

**Time Complexity: **O(N), where N is the index of nth multiple of k**Auxiliary Space: **O(1)

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