The least common multiple (LCM) of two or more numbers is the smallest number that all of them divide into exactly. To find it, list the multiples of each number and pick the first one they share, or multiply every prime factor at its highest power. For example, the LCM of 4 and 6 is 12.
In short: Small numbers: list multiples until one appears in every list. Bigger numbers: multiply each prime at the highest power it appears. Two numbers: LCM = (a × b) ÷ GCF.
In this guide
Method 1: List the Multiples
A multiple is what you get when you multiply a number by 1, 2, 3 and so on. Write the multiples of each number until you find one that is in every list.
Example: LCM of 4 and 6. Multiples of 4: 4, 8, 12, 16, 20, 24. Multiples of 6: 6, 12, 18, 24. The first common one is 12. (24 is also common, but it is not the least.)
A quick tip: list multiples of the bigger number only, and stop at the first one the smaller number divides into. For 9 and 12: 12, 24, 36. 36 ÷ 9 = 4, so the LCM is 36.
Method 2: Prime Factorization
- Write each number as a product of primes.
- Take every prime that appears, using the highest power it has in any one number.
- Multiply them. The product is the LCM.
Example: LCM of 12 and 18. 12 = 2² × 3 and 18 = 2 × 3². The highest powers are 2² and 3², so the LCM is 4 × 9 = 36.
The ladder method gives the LCM too: multiply the primes down the left side by the numbers left at the bottom. For 48 and 180 that is 2 × 2 × 3 × 4 × 15 = 720.
Method 3: Use the GCF (Two Numbers)
For two numbers, the GCF and LCM are linked: LCM = (a × b) ÷ GCF. If you already know the GCF, this is the fastest way.
Example: LCM of 48 and 180. The GCF is 12, so the LCM = 48 × 180 ÷ 12 = 8,640 ÷ 12 = 720.
LCM of Three or More Numbers
Example: LCM of 8, 12 and 16. 8 = 2³, 12 = 2² × 3 and 16 = 2⁴. The highest powers are 2⁴ and 3, so the LCM is 16 × 3 = 48.
Example: LCM of 15, 20 and 25. 15 = 3 × 5, 20 = 2² × 5 and 25 = 5². Multiply 2² × 3 × 5² = 4 × 3 × 25 = 300.
You can also work in pairs: LCM(8, 12) = 24, then LCM(24, 16) = 48.
The LCM and the Least Common Denominator
When you add or subtract fractions with different denominators, you need a common denominator. The smallest one, the least common denominator (LCD), is the LCM of the denominators.
Example: 1/4 + 5/6. The LCM of 4 and 6 is 12, so 1/4 = 3/12 and 5/6 = 10/12. The sum is 13/12 = 1 1/12. See how to add fractions, or let the fraction calculator find the LCD for you.
Try it
Find the LCM of up to 10 numbers, or the LCD of fractions, with every step.
Common Examples
| Numbers | LCM | How |
|---|---|---|
| 4, 6 | 12 | first common multiple |
| 6, 8 | 24 | 2³ × 3 |
| 9, 12 | 36 | 2² × 3² |
| 12, 18 | 36 | 2² × 3² |
| 8, 12, 16 | 48 | 2⁴ × 3 |
| 15, 20, 25 | 300 | 2² × 3 × 5² |
| 48, 180 | 720 | 48 × 180 ÷ 12 |
FAQ
What is the least common multiple?
The least common multiple (LCM) is the smallest positive number that each of the given numbers divides into exactly. The LCM of 4 and 6 is 12.
How do you find the LCM of two numbers?
List multiples of the larger number until the smaller one divides into one, or multiply every prime at its highest power. You can also use LCM = (a × b) ÷ GCF.
What is the LCM of 12 and 18?
The LCM of 12 and 18 is 36, because 12 = 2² × 3, 18 = 2 × 3², and 2² × 3² = 36.
What is the LCM of two prime numbers?
Two different primes share no factor, so their LCM is their product. The LCM of 7 and 13 is 91.
Is the LCM the same as the least common denominator?
The least common denominator of some fractions is the LCM of their denominators. For 1/4 and 5/6 it is 12.
Sources and Further Reading
- OpenStax, Prealgebra, Section 2.5: Prime Factorization and the Least Common Multiple, Rice University.
- Common Core Grade 6: The Number System (6.NS.B.4).
Next, compare the two in GCF vs LCM or review how to find the GCF. Check any answer with the GCF and LCM calculator.
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