Common Multiples – Multiples That Numbers Share
A common multiple of two or more numbers is a number that appears in the multiple lists of all of them. Just as common factors are shared, common multiples are shared destinations on the number line.
How to Find Common Multiples
List the multiples of each number. Identify numbers that appear in every list.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48...
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...
Common multiples: 12, 24, 36, 48, 60... (they continue infinitely)
Multiples of 3: 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60...
Multiples of 4: 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60...
Multiples of 5: 5,10,15,20,25,30,35,40,45,50,55,60...
First common multiple: 60
There Are Infinitely Many
Once you find one common multiple, you can find them all by multiplying it by 1, 2, 3, ... For example, since 12 is a common multiple of 4 and 6, so are 24, 36, 48, ...
Why Common Multiples Matter
| Application | Connection to Common Multiples |
|---|---|
| Adding fractions (1/4 + 1/6) | Need a common denominator — a common multiple of 4 and 6 |
| Scheduling (bus A every 4 min, bus B every 6 min) | Next coincidence at LCM = 12 minutes |
| Tiling problems | Tile dimensions must be common multiples |
Key Takeaways
- Common multiples appear in the multiple lists of all given numbers.
- There are infinitely many common multiples of any set of numbers.
- The smallest common multiple is the LCM.
- All other common multiples are multiples of the LCM.
Practice Questions
- List the first three common multiples of 5 and 8.
- List the first three common multiples of 6 and 9.
- Are 30, 60 and 90 all common multiples of 3 and 10? Verify.
- Find the first common multiple of 4, 6 and 8.
- Bus A runs every 8 minutes, Bus B every 10 minutes. They both depart at noon. When is the next time they depart together?