FactorsLCM: lowest common multiple, three methods
The lowest common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of them. It is also called the least common multiple.
Method 1: listing multiples
Good for small numbers. Find the LCM of 6 and 8.
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48
- Multiples of 8: 8, 16, 24, 32, 40, 48
The first number in both lists is 24, so LCM(6, 8) = 24. A quicker form: go through the multiples of the larger number, 8, 16, 24, and stop at the first one the smaller number divides.
Method 2: prime factorisation
Write each number as a product of primes. Take every prime that appears in any of the numbers, each with its largest power, and multiply.
- 12 = 2² × 3
- 18 = 2 × 3²
The largest power of 2 is 2², and of 3 is 3². LCM = 2² × 3² = 36.
Method 3: common division
This is the usual school method for three or more numbers. Write the numbers in a row. Divide by a prime that divides at least one of them, and bring down any number it does not divide. Repeat until the row is all 1s. Multiply the divisors.
| 2 | 12 | 18 | 30 |
|---|---|---|---|
| 3 | 6 | 9 | 15 |
| 2 | 2 | 3 | 5 |
| 3 | 1 | 3 | 5 |
| 5 | 1 | 1 | 5 |
| 1 | 1 | 1 |
LCM(12, 18, 30) = 2 × 3 × 2 × 3 × 5 = 180.
Using the HCF
For two numbers, LCM = product of the numbers ÷ HCF. For 24 and 36 the HCF is 12, so the LCM is 24 × 36 ÷ 12 = 72. This shortcut is for two numbers only. It does not work directly for three.
Facts about LCM
- The LCM is never smaller than the largest of the numbers.
- If one number divides the other, the larger number is the LCM: LCM(8, 32) = 32.
- If the numbers are co-prime, the LCM is their product: LCM(8, 15) = 120.
- Every common multiple of the numbers is a multiple of their LCM.
When a question needs the LCM
Look for the words smallest, least, minimum, or events that repeat and must happen together again.
- Bells ring every 6 and 8 minutes. They ring together every 24 minutes.
- The smallest number that leaves no remainder when divided by 12, 18 and 30 is 180.
- Adding fractions: the LCM of the denominators is the common denominator. For 1/12 + 1/18 use 36.
A common slip
Multiplying the numbers together always gives a common multiple, but rarely the lowest. 12 × 18 = 216 is a common multiple of 12 and 18, yet the LCM is 36. Using 216 as a common denominator is not wrong, but it makes the sum longer and the answer needs simplifying.
More solved questions are on the word problems page, and the calculator shows all the working for your own numbers.