Smallest Common Multiple
Problem Explanation
The smallest common multiple between two numbers is the smallest number that both numbers can divide into evenly. This concept can be extended to more than two numbers as well.
We can first start with finding the smallest common multiple between two numbers. Naively, we can start writing out multiple of each number until we write a multiple that exists from both numbers.
An example would be the numbers 3 and 4. The multiples of 3 are 3, 6, 9, 12, 15, 18, ... and the multiples of 4 are 4, 8, 12, 16, 20, .... The first smallest number we run into in both lists is 12 so this is the smallest common multiple between 3 and 4.
An faster approach is to check all multiples of 4 to see if they are also multiples of 3, by checking the remainder when we divide the multiple of 4 by 3.
Be careful - do not forget the keyword range. If we are given [1, 5], then we have to check for the smallest common multiple for all the numbers [1, 2, 3, 4, 5], which is the smallest number that is evenly divisible by all of them.
Relevant Links
Hints
Hint 1
You can use remainder operator (%) to check if the reminder of a division is 0, which means it is evenly divisible.
Solutions
Solution 1 - Looping approach (Click to Show/Hide)
function smallestCommons(arr) {
// Setup
const [min, max] = arr.sort((a, b) => a - b);
const numberDivisors = max - min + 1;
// Largest possible value for SCM
let upperBound = 1;
for (let i = min; i <= max; i++) {
upperBound *= i;
}
// Test all multiples of 'max'
for (let multiple = max; multiple <= upperBound; multiple += max) {
// Check if every value in range divides 'multiple'
let divisorCount = 0;
for (let i = min; i <= max; i++) {
// Count divisors
if (multiple % i === 0) {
divisorCount += 1;
}
}
if (divisorCount === numberDivisors) {
return multiple;
}
}
}
smallestCommons([1, 5]);
In this solution, we check every multiple of the largest value in the range until we find a value that is divisible by every value in the range.
The upper bound for this loop is the product of all values in the provided range, because this number will be divisible by every value in the range.
Solution 2 - ES6 looping (Click to Show/Hide)
function smallestCommons(arr) {
// Setup
const [min, max] = arr.sort((a, b) => a - b);
const range = Array(max - min + 1)
.fill(0)
.map((_, i) => i + min);
// Largest possible value for SCM
const upperBound = range.reduce((prod, curr) => prod * curr);
// Test all multiples of 'max'
for (let multiple = max; multiple <= upperBound; multiple += max) {
// Check if every value in range divides 'multiple'
const divisible = range.every((value) => multiple % value === 0);
if (divisible) {
return multiple;
}
}
}
smallestCommons([1, 5]);
This solution uses ES6 syntax to condense the logic in Solution 1.
Solution 3 - GCD and LCM (Click to Show/Hide)
function smallestCommons(arr) {
// Setup
const [min, max] = arr.sort((a, b) => a - b);
const range = Array(max - min + 1)
.fill(0)
.map((_, i) => i + min);
// GCD of two numbers
// https://en.wikipedia.org/wiki/Greatest_common_divisor#Euclid's_algorithm
const gcd = (a, b) => (b === 0) ? a : gcd(b, a % b);
// LCM of two numbers
// https://en.wikipedia.org/wiki/Least_common_multiple#Using_the_greatest_common_divisor
const lcm = (a, b) => a * b / gcd(a, b);
// LCM of multiple numbers
// https://en.wikipedia.org/wiki/Least_common_multiple#Other
return range.reduce((multiple, curr) => lcm(multiple, curr));
}
smallestCommons([1, 5]);
This solution uses formulae and algorithms for the Greatest Common Divisor and Least Common Multiple off of Wikipedia to compactly and quickly compute the Smallest Common Multiple.
Solution 4 - Prime factorization (Click to Show/Hide)
// Find the SCM of a range of numbers
function smallestCommons(arr) {
const primeFactors = {};
const [min, max] = arr.sort((a, b) => a - b);
for (let i = min; i <= max; i++) {
// Factorize number in range
const currentFactors = getPrimeFactors(i);
for (let prime in currentFactors) {
// Add factor to set or update number of occurrences
if (!primeFactors[prime] || currentFactors[prime] > primeFactors[prime]) {
primeFactors[prime] = currentFactors[prime]
}
}
}
// Build SCM from factorization
let multiple = 1;
for (let prime in primeFactors) {
multiple *= prime ** primeFactors[prime];
}
return multiple;
}
// Compute prime factors of a number
function getPrimeFactors(num) {
const factors = {};
for (let i = 2; i <= num; i++) {
// Count occurances of factor
// Note that composite values will not divide num
while ((num % i) === 0) {
factors[i] = (factors[i]) ? factors[i] + 1 : 1;
num /= i;
}
}
return factors;
}
smallestCommons([1, 5]);
This solution uses a prime factorization of the numbers in the range to compute the smallest common multiple. In general, Solution 3 is much faster, but with a very large range or very large values, sometimes Solution 3 may trigger a recursion limit in some browsers.