Least Common Multiple (LCM)

Learn how to find the smallest number that is a multiple of two or more numbers.

Intermediate25 minLesson
Printable worksheet

Definition

The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is a multiple of all the given numbers.
For example, to find the LCM of 44 and 66:
  • Multiples of 44: 4,8,12,16,20,24,...4, 8, 12, 16, 20, 24, ...
  • Multiples of 66: 6,12,18,24,30,...6, 12, 18, 24, 30, ...
  • Common multiples: 12,24,36,...12, 24, 36, ...
  • **LCM = 1212** (the smallest common multiple)
We write this as: LCM(4,6)=12\text{LCM}(4, 6) = 12

Try it now

What is the LCM of 22 and 33?

Worked Examples

Find the LCM of 33 and 55.
1

List multiples of 3

3×1=33 \times 1 = 3, 3×2=63 \times 2 = 6, 3×3=93 \times 3 = 9, 3×4=123 \times 4 = 12, 3×5=153 \times 5 = 15, ... → 3,6,9,12,15,18,21,...3, 6, 9, 12, 15, 18, 21, ...
2

List multiples of 5

5×1=55 \times 1 = 5, 5×2=105 \times 2 = 10, 5×3=155 \times 3 = 15, ... → 5,10,15,20,25,...5, 10, 15, 20, 25, ...
3

Find the smallest common multiple

Both lists contain 1515 → LCM(3,5)=15\text{LCM}(3, 5) = 15

Common Mistakes

Multiplying the numbers instead of finding LCM

Why it's wrong: While 4×6=244 \times 6 = 24 gives a common multiple, it is not always the least. The LCM of 44 and 66 is 1212, not 2424.

Correct: Always check if there is a smaller common multiple by listing or using prime factorization.

Confusing LCM with GCF

Why it's wrong: LCM is the smallest common multiple; GCF is the largest common factor. For 44 and 66: LCM is 1212, GCF is 22.

Correct: Remember: LCM uses multiples (bigger numbers), GCF uses factors (smaller numbers).

Missing a common multiple when listing

Why it's wrong: Students sometimes stop listing too early and pick a number that is not the smallest.

Correct: Keep listing until you find a number that appears in both lists. The first such number is the LCM.

Interactive Visual

Venn Diagram - GCF & LCM

Number A:
Number B:
46223Only in ACommonOnly in B

Prime factors of 4:

22

Prime factors of 6:

23

GCF

2

2 = 2

LCM

12

2 × 2 × 3 = 12

How it works:

  • • GCF = product of common prime factors (center of diagram)
  • • LCM = product of all prime factors (entire diagram)

Find the least common multiple using prime factorization.

Factor Tree

Number:
Composite
Prime

Enter a number to see its factor tree and prime factorization.

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

15 problems
Problem 1 of 15
Easy
What is the LCM of 22 and 33?

Why It Matters

The LCM is essential in many mathematical and real-world situations:
  • Adding fractions: To add 14+16\frac{1}{4} + \frac{1}{6}, you need LCM(4,6)=12\text{LCM}(4, 6) = 12 as the common denominator
  • Scheduling: If Bus A comes every 4 minutes and Bus B every 6 minutes, they arrive together every 12 minutes
  • Packaging: Pencils come in packs of 4 and erasers in packs of 6. The smallest equal number of each you can buy is LCM(4,6)=12\text{LCM}(4, 6) = 12 (3 packs and 2 packs)
  • Music: A drum pattern of 3 beats and one of 4 beats start together again every LCM(3,4)=12\text{LCM}(3, 4) = 12 beats

Real World Applications

Bus Schedules

Use LCM to figure out when two buses on different schedules will arrive at the same time.

Example:

Bus A arrives every 88 minutes and Bus B arrives every 1212 minutes. If both arrive at 9:00 AM, when will they next arrive together?

1Try It Yourself

Bus A arrives every 88 minutes and Bus B every 1212 minutes. Both arrive now.

In how many minutes will they both arrive together again?

Step 1: Write the mathematical expression

Find LCM(8,12)\text{LCM}(8, 12)

Adding Fractions

LCM helps find the common denominator when adding fractions with different denominators.

Example:

To add 14+16\frac{1}{4} + \frac{1}{6}, find LCM(4,6)=12\text{LCM}(4, 6) = 12, then rewrite as 312+212=512\frac{3}{12} + \frac{2}{12} = \frac{5}{12}.

2Try It Yourself

You want to add 13+15\frac{1}{3} + \frac{1}{5}.

What common denominator should you use?

Step 1: Write the mathematical expression

Find LCM(3,5)\text{LCM}(3, 5)

Key Takeaways

  • 1The LCM is the smallest positive number that is a multiple of all given numbers
  • 2Method 1: List multiples of each number and find the smallest common one
  • 3Method 2: Use prime factorization and take the highest power of each prime
  • 4LCM is used for finding common denominators when adding fractions
  • 5Do not confuse LCM (multiples, larger) with GCF (factors, smaller)

Frequently Asked Questions

LCM (Least Common Multiple) is the smallest number that both numbers divide into evenly. GCF (Greatest Common Factor) is the largest number that divides into both numbers evenly. LCM is always greater than or equal to the larger number; GCF is always less than or equal to the smaller number.
LCM (Least Common Multiple) is the smallest number that both numbers divide into evenly. GCF (Greatest Common Factor) is the largest number that divides into both numbers evenly. LCM is always greater than or equal to the larger number; GCF is always less than or equal to the smaller number.
Yes! If one number is a multiple of the other, the LCM equals the larger number. For example, LCM(3,6)=6\text{LCM}(3, 6) = 6 because 66 is already a multiple of 33.
Yes! You can use the formula: LCM(a,b)=a×bGCF(a,b)\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}. For example, for 44 and 66: GCF is 22, so LCM =4×62=242=12= \frac{4 \times 6}{2} = \frac{24}{2} = 12.

Glossary

Multiple
A number that can be divided evenly by another number. For example, 1212 is a multiple of 44 because 12÷4=312 \div 4 = 3.
Common multiple
A number that is a multiple of two or more numbers. For example, 1212 is a common multiple of 33 and 44.
Least Common Multiple (LCM)
The smallest positive common multiple of two or more numbers.
Prime factorization
Writing a number as a product of its prime factors. For example, 12=22×312 = 2^2 \times 3.
Today's challengeOne problem, three tries

More in This Topic

Factors and Multiples

Elementary20 min

Prime Factorization

Intermediate25 min

Prime and Composite Numbers

Elementary20 min

Greatest Common Factor (GCF)

Intermediate25 min