Least common multiple calculator

This least common multiple calculator (LCM) helps you quickly find the smallest positive number that is evenly divisible by two or more integers.

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What Is the least common multiple (LCM)?

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all of the numbers. That means the LCM is the first number that appears in every number’s list of multiples.

For example:

• The multiples of 3 are: 3, 6, 9, 12, …
• The multiples of 4 are: 4, 8, 12, 16, …
• The first number that appears in both lists is 12, so 12 is the LCM of 3 and 4.

Notation

Mathematically, the least common multiple of numbers (a) and (b) is written as: LCM(a, b)

If you have more than two numbers, you can extend this notation, e.g.: LCM(a, b, c)

The LCM always refers to the smallest positive common multiple.

How to find the LCM

There are several standard methods to compute the LCM:

1. Listing multiples

List out the multiples of each number until you find the smallest number they all share.

Example:
Find the LCM of 6 and 8.

• Multiples of 6:
  6, 12, 18, 24, 30, …
• Multiples of 8:
  8, 16, 24, 32, …

The smallest number that appears in both lists is 24.

LCM(6, 8) = 24

2. Using greatest common divisor

The LCM of two numbers can be calculated using their greatest common divisor (GCD) with the formula:

\text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)}

This allows you to compute the LCM using the GCD if you already have it.

Example:
Find the LCM of 20 and 30.

• GCD(20, 30) = 10
• Multiply the numbers: 20×30=600
• Divide by the GCD: 600÷10=60

LCM(20, 30) = 60

3. Prime factorization

• Break each number down into prime factors.
• The LCM is created by taking each prime factor at its highest power across all numbers.

Example:
Find the LCM of 12 and 18.

Prime factorizations:

• 12=2^2 \times 3^1
• 18 = 2^1 \times 3^2

Take the highest power of each prime:

• (2^2) \text{ and } (3^2)

LCM: (2^2) \text{ and } (3^2) = 4 \times 9 = 36

LCM(12, 18) = 36

4. LCM of more than two numbers

To find the LCM of three or more numbers, you can apply the LCM operation step by step.

Example:
Find the LCM of 4, 6, and 10.

1. LCM(4, 6) = 12
2. LCM(12, 10) = 60

LCM(4, 6, 10) = 60

5. Special cases

• Identical numbers:
  LCM(9, 9) = 9
• One number divides another:
  LCM(15, 45) = 45
• Coprime numbers (no common factors):
  LCM(7, 9) = $7 \times 9 = 63$7×9=63

Key properties of LCM

Here are some important features of the LCM: /

• Commutative:
  \text{LCM}(a, b) = \text{LCM}(b, a)
  Order doesn’t matter.
• Associative:
  \text{LCM}(a, b, c) = \text{LCM}(\text{LCM}(a, b), c)
  This makes it easy to compute LCM for more than two numbers in steps.
• LCM of a Number and Itself:
  \text{LCM}(a, a) = a
  The LCM of identical numbers is the number itself.
• LCM of Coprime Numbers:
  If two numbers share no factors (their GCD is 1), then:
  \text{LCM}(a, b) = a × b
  
• Multiples Relationship:
  If one number is a multiple of another, then the LCM is the larger number.
  Example: LCM(25, 75) = 75.

Uses of LCM

Fractions

LCM is used to find the lowest common denominator when adding, subtracting, or comparing fractions.

Scheduling and events

LCM helps determine when repeating events coincide — for example, if one event recurs every 6 minutes and another every 8 minutes, the LCM tells you when they will next happen at the same time.

Engineering and mechanics

In mechanical systems with gears or periodic cycles, LCM can indicate when systems will realign or repeat their patterns.