Least Common Multiple Calculator

What this calculator does

The Least Common Multiple Calculator finds the smallest positive whole number that is divisible by every number in a list. It also shows the prime factorization of each input so you can see how the result is built.

The least common multiple, usually written LCM, is used to find common denominators when adding or subtracting fractions, to align repeating events, and to solve divisibility problems. For two numbers, the LCM is related to the greatest common factor by the equation LCM(a, b) = a × b ÷ GCF(a, b).

How to use this calculator

1. Enter two or more whole numbers separated by commas, spaces, or semicolons.
2. Click Calculate.
3. Read the LCM and the prime-factorization steps.

The default example is 330, 75, 450, 225. The least common multiple is 4,950.

How the least common multiple is calculated

The calculator uses prime factorization. Each number is written as a product of primes, and the LCM takes the highest power of every prime that appears in any input. For the default example:

330 = 2 × 3 × 5 × 11
75  = 3 × 5 × 5
450 = 2 × 3 × 3 × 5 × 5
225 = 3 × 3 × 5 × 5

LCM = 2 × 3 × 3 × 5 × 5 × 11 = 4,950

For two numbers, the calculator can also combine the greatest common factor with the product rule. For a longer list, it applies the two-number rule repeatedly.

What the results mean

The result is the smallest number that every input divides evenly. In the default example, 4,950 ÷ 330 = 15, 4,950 ÷ 75 = 66, 4,950 ÷ 450 = 11, and 4,950 ÷ 225 = 22. All four divisions are exact.

A common multiple is any number divisible by all inputs. The least common multiple is the smallest positive one. Multiples of the LCM are also common multiples, so the answer describes an entire family of numbers.

Assumptions and limitations

• Enter between two and ten positive whole numbers.
• Each number must be 1,000,000,000 or less.
• The result must fit in a safe JavaScript integer. If the exact LCM is too large, the calculator reports an error instead of returning an approximation.
• Duplicate values are allowed; they do not change the result.
• The calculator shows prime-factorization steps, but it does not draw a Venn diagram or a factor tree.

Frequently asked questions

What is the difference between LCM and GCF?

The LCM is the smallest number that every input divides evenly. The GCF is the largest number that divides every input evenly. For 12 and 18, the LCM is 36 and the GCF is 6.

Can I find the LCM of more than two numbers?

Yes. Enter up to ten numbers. The calculator combines them two at a time, which gives the same result as factoring the whole list at once.

Why is the LCM of two numbers not always their product?

The product counts shared factors twice. For example, 4 and 6 have a product of 24, but their LCM is 12 because 4 and 6 share a factor of 2. Dividing the product by the GCF removes the duplication.

What happens if one number is a multiple of another?

The larger number is the LCM. For example, the LCM of 3 and 12 is 12 because 12 is already divisible by 3.

Source

Reference: Calculator.net — Least Common Multiple Calculator.