nth Multiple of a Number in Fibonacci Series in JavaScript

The Fibonacci series is a sequence of numbers where each number is the sum of the two previous ones, usually starting with 0 and 1. Given two integers n and k, the task is to find the position of the nth multiple of k in the Fibonacci series. For instance, if k = 2 and n = 3, the output would be 9 since the third multiple of 2 in the Fibonacci series is 34, which appears at position 9. There are multiple ways to find the nth multiple of a number in the Fibonacci series in Javascript which is as follows:

Mathematical approach

The Fibonacci series has a repeating pattern. We can use this pattern to quickly figure out where the nth multiple of a number k is in the series by using a specific math method to recognize when a Fibonacci number is a multiple of k.

Example: The Position of nth Multiple of a Number in the Fibonacci Series using an Iterative Approach in JavaScript.

// JavaScript function to find the position of 
// nth multiple of a number k in the Fibonacci Series

function findPosition(k, n) {
  let f1 = 0;
  let f2 = 1;
  let i = 2;

  while (true) {
    let f3 = f1 + f2;
    f1 = f2;
    f2 = f3;

    if (f2 % k === 0) {
      return n * i;
    }

    i++;
  }
}

let n = 5;
let k = 4;

console.log(
  "Position of nth multiple of k in Fibonacci Series is",
  findPosition(k, n)
);

Position of nth multiple of k in Fibonacci Series is 30

Time Complexity: O(N)

Space Complexity: O(1)

Iterative Approach using a List

Another approach involves generating the Fibonacci series and checking for multiples of k. This function iteratively generates the Fibonacci series and checks for multiples of k.

Example: Finding the Position of nth Multiple of a Number in the Fibonacci Series using Iterative Approach in JavaScript.

// JavaScript function to find the position 
// of nth multiple of a number k in the 
// Fibonacci Series using an iterative approach

function nthFibMultiple(n, k) {
    let fibonacci = [0, 1];
    let counter = 0;

    for (let i = 2; ; i++) {
        let current = fibonacci[i - 1] + fibonacci[i - 2];

        if (current % k === 0) {
            counter++;

            if (counter === n) {
                return i;
            }
        }

        fibonacci.push(current);
    }
}

console.log(nthFibMultiple(3, 2));

9

Time Complexity: O(N)

Space Complexity: O(N)

Recursive Fibonacci Series with Memoization

Create a function "fib" that find the nth number in the Fibonacci series. while keeping a record of previously calculated Fibonacci numbers using memorization. Create another function, "fibMultiple(k, n)", to find the nth multiple of k in the Fibonacci series.

Example: Finding the Position of nth Multiple of a Number in the Fibonacci Series using Recursion and Memoization in JavaScript

// JavaScript function to find the position of 
// nth multiple of a number k in the Fibonacci 
// Series using recursion and memoization

function fib(n, memo) {
  if (n === 1 || n === 2) {
    return 1;
  }

  if (memo[n] !== 0) {
    return memo[n];
  }

  memo[n] = fib(n - 1, memo) + fib(n - 2, memo);
  return memo[n];
}

function fibMultiple(k, n) {
  let memo = Array(100).fill(0);
  let count = 0;

  for (let i = 1; ; i++) {
    if (fib(i, memo) % k === 0) {
      count++;

      if (count === n) {
        return i;
      }
    }
  }
}

let k = 2;
let n = 3;
let result = fibMultiple(k, n);
console.log(result);

9

Time Complexity: O(N)

Space Complexity: O(N)