Least Common Multiple (LCM)
Learn how to find the smallest number that is a multiple of two or more numbers.
Definition
- Multiples of :
- Multiples of :
- Common multiples:
- **LCM = ** (the smallest common multiple)
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Worked Examples
Find the LCM of and .
List multiples of 3
, , , , , ... →
List multiples of 5
, , , ... →
Find the smallest common multiple
Both lists contain →
Answer: The LCM of and is .
Common Mistakes
Multiplying the numbers instead of finding LCM
Why it's wrong: While gives a common multiple, it is not always the least. The LCM of and is , not .
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 and : LCM is , GCF is .
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: Continue listing until you find a match in both lists. The first match is the LCM.
Interactive Visual
Venn Diagram - GCF & LCM
Prime factors of 4:
Prime factors of 6:
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
Enter a number to see its factor tree and prime factorization.
Interactive Sandbox
Expression Calculator
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Practice Problems
15 problemsWhat is the LCM of and ?
Why It Matters
- Adding fractions: To add , you need 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: A factory needs boxes that fit products in groups of 4 and 6 perfectly
- Music: Finding when rhythms of different beats align together
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 minutes and Bus B arrives every minutes. If both arrive at 9:00 AM, when will they next arrive together?
Bus A arrives every minutes and Bus B every minutes. Both arrive now.
In how many minutes will they both arrive together again?
Step 1: Write the mathematical expression
Find
Adding Fractions
LCM helps find the common denominator when adding fractions with different denominators.
Example:
To add , find , then rewrite as .
You want to add .
What common denominator should you use?
Step 1: Write the mathematical expression
Find
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
Glossary
- Multiple
- A number that can be divided evenly by another number. For example, is a multiple of because .
- Common multiple
- A number that is a multiple of two or more numbers. For example, is a common multiple of and .
- 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, .