JavaScript Program to Find Smallest Number that is Divisible by First n Numbers

Last Updated : 24 Jun, 2024

Given a number, our task is to find the smallest number that is evenly divisible by the first n numbers without giving any remainder in JavaScript.

Example:

Input: 10
Output: 2520 
Explanation: 
2520 is smallest number which is completely divisible by numbers from 1 to 10.

Below are the following approaches for finding the Smallest number that is divisible by the first n numbers:

Table of Content

Brute Force Approach
Using LCM (Lowest Common Multiple)
Using Prime Factorization Method:
Using Array.from and Recursion

Brute Force Approach

To Find the smallest number divisible by the first n numbers using Brute Force Approach first initialize num with n. Use a while loop to find the smallest divisible number. In the loop, set divisible to true, and iterate from 1 to n, checking if num is divisible by i. If not, set divisible to false and break. If divisible remains true, return num. Otherwise, increment num and continue.

Example: Demonstration of Finding the Smallest number divisible by the first n numbers using the Brute Force Approach.

JavaScript

function smallDivi(n) {
    let num = n;
    while (true) {
        let divisible = true;
        for (let i = 1; i <= n; i++) {
            if (num % i !== 0) {
                divisible = false;
                break;
            }
        }
        if (divisible) {
            return num;
        }
        num++;
    }
}
console.log(smallDivi(10));

Output

Time Complexity: O(num * n)

Space Complexity: O(1)

Using LCM (Lowest Common Multiple)

In this approach, first implement gcd to find the greatest common divisor using Euclid's algorithm. Implement lcm to calculate the least common multiple as a * b / gcd(a, b). Initialize the answer to 1 and iterate from 2 to n, updating the answer with the least common multiple of the answer and i. After the loop, the answer holds the smallest number divisible by the first n numbers. Return answer.

Example: Demonstration of Finding the the Smallest number that is divisible by first n numbers using LCM (Lowest Common Multiple).

JavaScript

function smallestDivisibleLCM(n) {
    function calculateGCD(a, b) {
        return b === 0 ? a : calculateGCD(b, a % b);
    }

    function calculateLCM(a, b) {
        return (a * b) / calculateGCD(a, b);
    }

    // Initialize the result to 1
    let answer = 1;

    // Iterate through numbers from 2 to n to calculate LCM 
    for (let i = 2; i <= n; i++) {
        answer = calculateLCM(answer, i);
    }

    return answer;
}

console.log(smallestDivisibleLCM(10));

Output

Time Complexity: O(n * log(n)).

Space Complexity: O(1).

Using Prime Factorization Method:

The Prime Factorization Method calculates the smallest number divisible by the first n numbers by identifying all primes up to n, determining the highest power of each prime that fits within n, and multiplying these highest powers together.

Example: In this example we will follow the above explained approach.

JavaScript

function sieve(n) {
    let isPrime = Array(n + 1).fill(true);
    isPrime[0] = isPrime[1] = false;
    for (let i = 2; i * i <= n; i++) {
        if (isPrime[i]) {
            for (let j = i * i; j <= n; j += i) {
                isPrime[j] = false;
            }
        }
    }
    return isPrime.map((prime, index) => prime ? index : false).filter(Boolean);
}

function highestPowerOfPrimeUnderN(prime, n) {
    let power = prime;
    while (power * prime <= n) {
        power *= prime;
    }
    return power;
}

function smallestDivisibleByFirstN(n) {
    const primes = sieve(n);
    let result = 1;

    primes.forEach(prime => {
        result *= highestPowerOfPrimeUnderN(prime, n);
    });

    return result;
}

let n = 10;
console.log(`The smallest number that is divisible by the first ${n} numbers is:
${smallestDivisibleByFirstN(n)}`);

Output

The smallest number that is divisible by the first 10 numbers is:
2520

Using Array.from and Recursion

This approach leverages recursion to compute the LCM of a list of numbers. By breaking down the problem into smaller subproblems, it recursively calculates the LCM of two numbers at a time.

Example: In this example, we will use a recursive approach to find the smallest number divisible by the first n numbers.

JavaScript

function calculateGCD(a, b) {
    return b === 0 ? a : calculateGCD(b, a % b);
}

function calculateLCM(a, b) {
    return (a * b) / calculateGCD(a, b);
}

function recursiveLCM(arr) {
    if (arr.length === 2) {
        return calculateLCM(arr[0], arr[1]);
    } else {
        const lastElement = arr.pop();
        return calculateLCM(lastElement, recursiveLCM(arr));
    }
}

function smallestDivisibleRecursiveLCM(n) {
    const numbers = Array.from({ length: n }, (_, i) => i + 1);
    return recursiveLCM(numbers);
}

console.log(smallestDivisibleRecursiveLCM(10));

Output

bug8wdqo

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Article Tags :

JavaScript Program to Find Smallest Number that is Divisible by First n Numbers

Brute Force Approach

Using LCM (Lowest Common Multiple)

Using Prime Factorization Method:

Using Array.from and Recursion

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