Integers
Lowest common multiples
List the multiples of two numbers and eventually they collide. The first place they meet is the lowest common multiple.
Multiples
The multiples of a number are what you get by multiplying it by 1, 2, 3, and so on. The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, … The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, …
Common multiples and the LCM
Numbers that appear in BOTH lists are common multiples. The common multiples of 4 and 6 are 12, 24, 36, … The smallest of them is the LOWEST COMMON MULTIPLE (LCM). The LCM of 4 and 6 is 12 — the smallest number that is a multiple of both.
A quicker way
You do not always need both full lists. To find the LCM of 6 and 10: The multiples of 6 are 6, 12, 18, 24, 30, 36, … A multiple of 10 ends in 0, and 30 is the first one in that list. So the LCM of 6 and 10 is 30.
A useful fact
For any two whole numbers A and B, A × B is always A common multiple of them — but it is not always the LOWEST one. 4 × 6 = 24, which is a common multiple. But the LCM is 12, which is smaller. When the two numbers share no factors, A × B IS the LCM: for 7 and 8, the LCM is 56.
Worked examples
The lowest common multiple of 4 and 6
- Multiples of 4: 4, 8, 12, 16, 20, 24, …
- Multiples of 6: 6, 12, 18, 24, …
- The first number in both lists is 12.
Answer: 12
The LCM of 6 and 10
- Multiples of 6: 6, 12, 18, 24, 30, …
- A multiple of 10 ends in 0, and 30 is the first such.
- So 30 is in both lists.
Answer: 30
The LCM of 7 and 8
- 7 and 8 share no factors.
- When that happens, the product IS the lowest common multiple.
- 7 × 8 = 56.
Answer: 56
Is 24 the LCM of 4 and 6?
- 24 is a common multiple of 4 and 6.
- But 12 is also common, and smaller.
Answer: No — 24 is common but not the lowest
The LCM of 3 and 5
- Multiples of 3: 3, 6, 9, 12, 15, …
- 15 is the first one that is also a multiple of 5.
Answer: 15
The LCM of 4 and 8
- 8 is itself a multiple of 4.
- So the smallest number in both lists is 8.
Answer: 8 — when one divides the other, the LCM is the larger one
Where you see this in real life
- Two buses leave a stop every 12 and 18 minutes. They next leave together after the LCM, 36 minutes.
- Packs of 8 burgers and 12 buns: to have none left over you need the LCM, 24 of each — 3 packs of burgers and 2 of buns.
Key words
- Multiple
- the result of multiplying a number by a whole number
- Common multiple
- a number that is a multiple of two or more numbers
- Lowest common multiple (LCM)
- the smallest common multiple of two numbers
- Product
- the result of multiplying numbers together
- Times table
- the list of multiples of a number