Least Common Multiple - Definition, Examples, Quiz, FAQ, Trivia

What is Least Common Multiple?

Visual explanation of multiples — Understanding multiples and common multiples

The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of each number. It's the smallest number that all the original numbers divide into without leaving a remainder.

Think of it as the first meeting point for multiples of different numbers. For example, the multiples of 3 are 3, 6, 9, 12, 15, 18... and the multiples of 4 are 4, 8, 12, 16, 20... The common multiples are 12, 24, 36... and the smallest is 12. So the LCM of 3 and 4 is 12.

LCM helps us solve problems where we need to find when things will happen at the same time or how to combine things evenly.

Methods to Find LCM

There are several ways to find the LCM of numbers. Let's explore three common methods:

Listing Multiples Method

List the multiples of each number until you find the smallest common multiple.

Example: Find LCM of 4 and 6

Multiples of 4: 4, 8, 12, 16, 20...

Multiples of 6: 6, 12, 18, 24, 30...

First common multiple is 12 → LCM = 12

Prime Factorization Method

Break each number into its prime factors and multiply the highest power of each prime.

Example: Find LCM of 12 and 18

12 = 2² × 3¹

18 = 2¹ × 3²

Highest powers: 2² × 3² = 4 × 9 = 36

LCM = 36

Division Method

Divide by common prime factors until no common factors remain, then multiply the divisors and remainders.

Example: Find LCM of 15 and 25

Divide by common factors:


5 | 15   25
   | --------
3 |  3    5
   | --------
5 |  1    5
   | --------
   |  1    1

LCM = 5 × 3 × 5 = 75

LCM of Co-prime Numbers

Visual explanation of co-prime numbers — Understanding co-prime numbers and their least common multiple

Co-prime numbers are numbers that have no common prime factors other than 1. This means their Greatest Common Divisor (GCD) is 1. Examples include 8 and 9, or 15 and 16.

For co-prime numbers, the LCM is simply their product. This is because they share no common factors, so we multiply all their prime factors together.

Co-prime LCM Formula

LCM(a, b) = a × b

When a and b are co-prime numbers

Example: Find LCM of 8 and 9
Since 8 and 9 are co-prime (no common factors except 1):
LCM(8, 9) = 8 × 9 = 72

This special relationship makes finding LCM for co-prime numbers very easy!

LCM in Fractions

Visual explanation of using LCM — Using LCM to find common denominators in fractions

LCM is especially useful when working with fractions. When adding or subtracting fractions with different denominators, we need to find a common denominator. The LCM of the denominators is the smallest possible common denominator, called the Lowest Common Denominator (LCD).

Why use LCM for fractions?

It gives the smallest common denominator, keeping numbers manageable
It simplifies the process of adding, subtracting, and comparing fractions
It helps reduce fractions to their simplest form

Example: Add 1/4 + 1/6
Step 1: Find LCM of denominators 4 and 6 → LCM(4,6) = 12
Step 2: Convert fractions: 1/4 = 3/12 and 1/6 = 2/12
Step 3: Add: 3/12 + 2/12 = 5/12

Using LCM as the common denominator makes fraction operations much easier!

LCM Practice Quiz

Test your understanding with this 5-question quiz. Choose the correct answer for each question.

Frequently Asked Questions

Here are answers to common questions about Least Common Multiple:

What's the difference between LCM and GCF?

Can LCM be smaller than the numbers?

How do I find LCM for more than two numbers?

Why is LCM important in real life?

Math Trivia

Discover interesting facts about multiples and mathematics:

Least Common Multiple - Definition, Examples, Quiz, FAQ, Trivia

What is Least Common Multiple?

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