Free Math Guide

Least Common Multiple Explained

Learn how to generate multiples in order, compare two lists, identify common multiples, and choose the smallest value shared by both numbers.

Practice Least Common Multiples Read the Guide

The least common multiple, or LCM, is the smallest positive number that is a multiple of two or more numbers. One of the clearest ways to find it is to list multiples of each number in order and identify the first value that appears in every list.

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The Basic Idea

What Is the Least Common Multiple?

The least common multiple is the smallest positive whole number that is a multiple of every number being compared.

LCM(6, 8) = 24

Common

The number must appear in the multiples of both original numbers.

Least

Of all the shared multiples, choose the smallest positive one.

Check the Result

Since 24 ÷ 6 = 4 and 24 ÷ 8 = 3, the number 24 is a multiple of both 6 and 8. No smaller positive number is a multiple of both.

Example: LCM of 4 and 10

Some common multiples of 4 and 10 are 20, 40, 60, and 80. The smallest is 20.

LCM(4, 10) = 20

Repeated Multiplication

What Is a Multiple?

A multiple is the result of multiplying a number by a whole number. To generate multiples in order, multiply the original number by 1, 2, 3, 4, and so on.

Multiples of 6

  1. 6 × 1 = 6
  2. 6 × 2 = 12
  3. 6 × 3 = 18
  4. 6 × 4 = 24
  5. 6 × 5 = 30
  6. 6 × 6 = 36
6 12 18 24 30 36

Multiples Continue Forever

A number has infinitely many multiples. The dots in a multiples list mean the pattern continues.

Multiples Are Not the Same as Factors

Factors divide into a number. Multiples are created by multiplying the number. For example, 3 is a factor of 12, while 12 is a multiple of 3.

Method 1

Find the LCM by Listing Multiples

Listing multiples is a clear way to understand what the least common multiple means. Write multiples of each number in order and stop when you find the first match.

1

List the First Number’s Multiples

Multiply the first number by 1, 2, 3, and so on.

2

List the Second Number’s Multiples

Generate the second list in the same order.

3

Find Values in Both Lists

Identify the numbers that appear in both multiples lists.

4

Choose the First Match

The smallest shared value is the least common multiple.

Worked Example: LCM of 6 and 8

Multiples of 6

6 12 18 24 30 48

Multiples of 8

8 16 24 32 40 48

Common multiples shown: 24 and 48

The first and smallest common multiple is 24.

LCM(6, 8) = 24

Do Not Choose Just Any Common Multiple

Both 24 and 48 are common multiples of 6 and 8, but only 24 is the least common multiple because it is smaller.

Find the First Match

Common Multiples Continue After the LCM

Two numbers usually share more than one multiple. After finding the LCM, later common multiples are multiples of the LCM itself.

Common Multiples of 6 and 8

24 48 72 96 120

These common multiples increase by 24 each time.

That happens because every common multiple shown is also a multiple of the LCM.

Why the First Match Matters

When the multiples are listed from least to greatest, the first number appearing in both lists must be the least common multiple.

A Useful Shortcut

When One Number Is a Multiple of the Other

If one of the original numbers is already a multiple of the other, the larger number is the least common multiple.

Example: LCM of 4 and 12

Multiples of 4

4 8 12 16 20

Multiples of 12

12 24 36 48 60

The first multiple of 12 is already in the multiples of 4.

Therefore, the larger number is the LCM.

LCM(4, 12) = 12

Quick Check

Ask whether the larger number divides evenly by the smaller number. If it does, the larger number is the least common multiple.

Method 2

Find the LCM Using Prime Factorization

Prime factorization is a more efficient method when the numbers are larger. Rewrite both numbers as products of prime factors, then include every prime needed to build both numbers.

1

Prime Factorize Both Numbers

Continue factoring until every factor is prime.

2

Compare Each Prime

Identify every prime that appears in either factorization.

3

Keep the Greatest Number of Copies

For each prime, use the largest number of copies found in either factorization.

4

Multiply the Selected Factors

Their product is the least common multiple.

Worked Example: LCM of 24 and 36

24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3

The prime 2 appears three times in 24 and twice in 36, so keep three copies of 2. The prime 3 appears once in 24 and twice in 36, so keep two copies of 3.

Prime factors needed: 2 × 2 × 2 × 3 × 3

Multiply them: 2 × 2 × 2 × 3 × 3 = 72

LCM(24, 36) = 72

Why This Method Works

The LCM must contain enough copies of every prime factor to be divisible by both original numbers. Taking the greatest number of copies guarantees that both numbers divide the final product evenly.

Repeated Prime Factors

Use the Greatest Exponent

Exponents provide a shorter way to count repeated prime factors. For each prime, use the larger exponent from the two factorizations.

Example: LCM of 48 and 60

48 = 2⁴ × 3
60 = 2² × 3 × 5
Prime 2
Compare 2⁴ and 2². Keep 2⁴.
Prime 3
Both numbers contain one copy. Keep 3.
Prime 5
Only 60 contains 5, so include 5.

LCM prime factorization: 2⁴ × 3 × 5

16 × 3 × 5 = 240

LCM(48, 60) = 240

Do Not Add the Exponents

To find the LCM, choose the larger exponent for each prime. Do not add the exponents from the two factorizations.

Practice Examples

Step-by-Step Least Common Multiple Examples

These examples use both listing multiples and prime factorization so you can see when each method is useful.

Example 1: Listing Multiples

LCM(5, 8)
  1. Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40
  2. Multiples of 8: 8, 16, 24, 32, 40
  3. The first number appearing in both lists is 40.
LCM = 40

Example 2: Listing Multiples

LCM(9, 12)
  1. Multiples of 9: 9, 18, 27, 36, 45, 54
  2. Multiples of 12: 12, 24, 36, 48, 60
  3. The first shared multiple is 36.
LCM = 36

Example 3: Prime Factorization

LCM(18, 30)
  1. 18 = 2 × 3 × 3
  2. 30 = 2 × 3 × 5
  3. Use two copies of 3, one copy of 2, and one copy of 5.
  4. 2 × 3 × 3 × 5 = 90
LCM = 90

Example 4: One Number Divides the Other

LCM(7, 21)
  1. 21 ÷ 7 = 3.
  2. Therefore, 21 is already a multiple of 7.
  3. The larger number is the least common multiple.
LCM = 21

Avoid These Errors

Common Least Common Multiple Mistakes

Mistake 1: Choosing a Common Multiple That Is Not the Least

Multiples of 6 and 8 include 24 and 48.

❌ Choosing 48
✔ Choose 24 because it is the smallest common multiple.

Mistake 2: Confusing Factors and Multiples

Listing 1, 2, 3, and 6 as multiples of 6
✔ Those are factors. Multiples of 6 are 6, 12, 18, 24, and so on.

Mistake 3: Stopping Before the First Match

If the current lists do not contain a shared number yet, continue generating multiples in order until a match appears.

Mistake 4: Using Too Few Prime Factors

24 = 2³ × 3
36 = 2² × 3²

❌ Using 2² × 3²
✔ Use 2³ × 3² because the LCM needs the greatest exponent for each prime.

Mistake 5: Adding Prime Exponents

2³ and 2² become 2⁵
✔ Keep the larger exponent, 2³.

Mistake 6: Assuming the LCM Is Always the Product

LCM(6, 8) = 6 × 8 = 48
✔ The actual LCM is 24. The product only works automatically when the numbers are coprime.

Real-World Uses

Where You'll Use the Least Common Multiple

The least common multiple appears throughout middle school mathematics and many real-world situations involving repeating patterns and equal groupings.

Adding and Subtracting Fractions

Find the least common denominator by first finding the least common multiple of the denominators.

Repeating Events

Determine when two repeating schedules will happen together again.

Pattern Problems

Compare repeating numerical patterns and locate the first value shared by both.

Algebra

Simplify rational expressions and solve equations involving different denominators.

Quick Review

Least Common Multiple Checklist

✓ List Multiples
Write multiples in order beginning with multiplication by 1.
✓ Compare the Lists
Look for numbers appearing in every list.
✓ Choose the Smallest
The first shared value is the least common multiple.
✓ Use Prime Factorization
For larger numbers, keep the greatest exponent of every prime factor.
✓ Verify Your Answer
The LCM should divide evenly by every original number.

Practice Finding Least Common Multiples

Work through interactive problems by listing multiples, identifying common multiples, and using prime factorization with immediate feedback after every step.

Open the Interactive LCM Tool

Continue Learning

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