Least Common Multiple Calculator

LCM of Two Numbers

Enter two integers to find their Least Common Multiple along with a step-by-step prime factorization breakdown.

First number

Second number

LCM of Multiple Numbers

Enter three or more integers separated by commas to find the LCM of the entire set.

Numbers

Separate each number with a comma

LCM and GCD Calculator

Find both the Least Common Multiple and the Greatest Common Divisor of two numbers at once, with the relationship between them shown.

First number

Second number

Repeating Cycle Calculator

Find when two repeating events will next occur at the same time. Enter how often each event repeats to find the first point they coincide.

Event A repeats every units

Event B repeats every units

Unit of time

What is the Least Common Multiple?

The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of those integers without leaving a remainder. It is also called the Lowest Common Multiple or Smallest Common Multiple.

LCM(4, 6) = 12, because 12 is the smallest number divisible by both 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20…
Multiples of 6: 6, 12, 18, 24…
First common multiple: 12

How to Find the LCM Using Prime Factorization

The prime factorization method breaks each number into its prime factors. The LCM is found by taking the highest power of each prime factor that appears in any of the numbers.

Step 1: Find the prime factorization of each number.
Step 2: For each prime factor, take the highest power that appears.
Step 3: Multiply all these prime powers together.

Example: LCM(12, 18)
12 = 2 x 2 x 3 = 2^2 x 3^1
18 = 2 x 3 x 3 = 2^1 x 3^2
LCM = 2^2 x 3^2 = 4 x 9 = 36

How to Find the LCM Using the GCD Formula

A fast method uses the relationship between LCM and the Greatest Common Divisor (GCD). The product of two numbers always equals the product of their LCM and GCD.

LCM(a, b) = (a x b) / GCD(a, b)

Example: LCM(12, 18)
GCD(12, 18) = 6
LCM = (12 x 18) / 6 = 216 / 6 = 36

LCM vs GCD

The LCM and GCD are closely related. The GCD is the largest number that divides both integers evenly, while the LCM is the smallest number that both integers divide evenly into. For any two positive integers a and b, their relationship is:

LCM(a, b) x GCD(a, b) = a x b
This means: LCM(a, b) = (a x b) / GCD(a, b)

Real-World Applications of LCM

The LCM is used in many practical situations. When adding fractions with different denominators, the LCM of the denominators gives the least common denominator. In scheduling, the LCM determines when two recurring events next coincide. For example, if one bus comes every 6 minutes and another every 10 minutes, they will both arrive together every LCM(6, 10) = 30 minutes.