# 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.
```

 `# Python Program to find position of n\'th multiple ` `# of a mumber 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 Seires is"``, findPosition(k,n)); ` ` `  `# Code contributed by Mohit Gupta_OMG `

Output:

```Position of n'th multiple of k inFibonacci Seires is 30
```

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

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