# C/C++ 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.

`// C++ program to find position of n'th multiple ` `// of a mumber k in Fibonacci Series ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` `const` `int` `MAX = 1000; ` ` ` `// Returns position of n'th multple of k in ` `// Fibonacci Series ` `int` `findPosition(` `int` `k, ` `int` `n) ` `{ ` ` ` `// Iterate through all fibonacci numbers ` ` ` `unsigned ` `long` `long` `int` `f1 = 0, f2 = 1, f3; ` ` ` `for` `(` `int` `i = 2; i <= MAX; i++) { ` ` ` `f3 = f1 + f2; ` ` ` `f1 = f2; ` ` ` `f2 = f3; ` ` ` ` ` `// Found first multiple of k at position i ` ` ` `if` `(f2 % k == 0) ` ` ` ` ` `// n'th multiple would be at position n*i ` ` ` `// using Periodic property of Fibonacci ` ` ` `// numbers under modulo. ` ` ` `return` `n * i; ` ` ` `} ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `int` `n = 5, k = 4; ` ` ` `cout << ` `"Position of n'th multiple of k"` ` ` `<< ` `" in Fibonacci Series is "` ` ` `<< findPosition(k, n) << endl; ` ` ` `return` `0; ` `} ` |

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**Output:**

Position of n'th multiple of k in Fibonacci Series is 30

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

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