How to find the lowest common multiple (LCM)
The multiples of a number are its times table. Count in 4s and in 6s, and both reach 12. That is a common multiple. The lowest common multiple (LCM) is the first one they both reach. In the United States it is called the least common multiple. It means the same thing.
The rule
The lowest common multiple of two numbers is the smallest number that both of them divide into exactly.
How to do it
- Write the two numbers side by side, and draw a line down their right-hand side.
- Find a prime that divides at least one of them. Write it on the right of the line.
- Divide the numbers it goes into. Copy any number it does not go into straight down.
- Keep going until both numbers are 1. Multiply the primes on the right. That is the LCM.
Worked examples
What is the LCM of 12 and 18?
| 12 | 18 | 2 |
| 6 | 9 | 2 |
| 3 | 9 | 3 |
| 1 | 3 | 3 |
| 1 | 1 |
LCM = 2 × 2 × 3 × 3 = 36. 2 does not go into 9, so the 9 is copied down.
What is the LCM of 8 and 12?
| 8 | 12 | 2 |
| 4 | 6 | 2 |
| 2 | 3 | 2 |
| 1 | 3 | 3 |
| 1 | 1 |
LCM = 2 × 2 × 2 × 3 = 24.
What is the LCM of 4 and 5?
No prime divides both 4 and 5.
When two numbers are coprime, the LCM is just the two multiplied: 4 × 5 = 20.
Another way, for small numbers
List the multiples of each number until the same one turns up in both lists.
Multiples of 12: 12, 24, 36, 48. Multiples of 18: 18, 36, 54. The first number in both lists is 36.
This is easy for small numbers and slow for big ones. That is what the ladder is for.
Using it to add fractions
To add fractions, the bottom numbers have to match. The smallest bottom number that works is the LCM of the two, and it is called the lowest common denominator.
To add 1/4 and 1/6, find the LCM of 4 and 6, which is 12. 1/4 is 3/12 and 1/6 is 2/12, so the answer is 5/12.
How to check your answer
Both numbers have to divide the LCM exactly: 36 ÷ 12 = 3 and 36 ÷ 18 = 2.
The LCM is never smaller than the bigger of the two numbers. If yours is, something has gone wrong.
One more check: the HCF times the LCM is the two numbers multiplied. For 12 and 18, 6 × 36 = 216, and 12 × 18 = 216 too.
Where mistakes happen
- Dividing a number the prime does not go into. It is copied straight down, unchanged.
- Just multiplying the two numbers. 12 × 18 = 216 is a common multiple, but not the lowest one. That only works when the numbers are coprime.
- Mixing it up with the HCF. The LCM is never smaller than the numbers. The HCF is never bigger.
Questions people ask
Is it lowest or least common multiple?
Both. Lowest is British, least is American, and LCM is short for either.
What if one number divides the other?
Then the bigger number is the LCM. The LCM of 6 and 18 is 18.
What is the lowest common denominator?
The LCM of the bottom numbers of two fractions. For 1/4 and 1/6 it is 12.
Is there a shortcut if I know the HCF?
Yes. Multiply the two numbers, then divide by the HCF: 12 × 18 = 216, and 216 ÷ 6 = 36.