Factors, Multiples & Primes
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.
Working out when repeating cycles line up again is a genuinely ancient problem. The Greek astronomer Meton of Athens noticed, around 432 BCE, that 235 lunar months come astonishingly close to 19 solar years — a near-common-multiple now called the Metonic cycle, which several calendars still in use today (including the Hebrew calendar) rely on to decide when to insert a leap month. It is exactly the same reasoning behind the bus-timetable and traffic-light problems later on this page, just scaled up from minutes to millennia.
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.