How to Find the LCM: 4 Methods Step by Step
How to find the LCM: to find the least common multiple, write each number as a product of primes, take every prime at its highest power, and multiply. LCM(12, 18) = 2² × 3² = 36. Three other methods give the same answer: listing multiples, the division (ladder) method, and the GCF formula.
The LCM Calculator shows every method for the numbers you enter. The sections below explain each method on the same numbers so the results can be compared. The lowest common multiple (UK and India wording) and the least common multiple (US wording) are the same thing.
Method 1: Listing multiples
List the multiples of each number until the same number appears in every list. That number is the LCM.
Listing multiples: LCM(12, 18)
Write out the multiples of each number until one multiple appears in every row. The first number that appears in all rows is the least common multiple.
36 is the first multiple shared by every number.
LCM(12, 18) = 36
Listing multiples suits numbers up to about 12 and is the method taught first in grade 6 (US) and class 5 (India).
Method 2: Prime factorization
Break each number into prime factorisation, take the highest power of each prime, and multiply.
Prime factorization: LCM(12, 15, 75)
Break each number into prime factors.
- 12 = 2 × 2 × 3 = 2² × 3
- 15 = 3 × 5
- 75 = 3 × 5 × 5 = 3 × 5²
Take the highest power of every prime that appears, then multiply them together.
| Prime | Highest power | From |
|---|---|---|
| 2 | 2² | 12 |
| 3 | 3 | 12, 15, 75 |
| 5 | 5² | 75 |
LCM = 2² × 3 × 5² = 300
LCM(12, 15, 75) = 300
Prime factorization works for any amount of numbers and any size. Full page: LCM by prime factorisation.
Method 3: Division (ladder) method
Write the numbers in a row, divide by the smallest prime that goes into at least one of them, carry the rest down, and repeat until every number is 1. The LCM is the product of the divisors. The division method / ladder method is the standard NCERT class 6 layout.
Division method: LCM(12, 15, 75)
Divide the row by the smallest prime that goes into at least one number; carry the rest down. Keep going until every number becomes 1. The LCM is the product of all the divisors on the left.
| 2 | 12 | 15 | 75 |
|---|---|---|---|
| 2 | 6 | 15 | 75 |
| 3 | 3 | 15 | 75 |
| 5 | 1 | 5 | 25 |
| 5 | 1 | 1 | 5 |
| 1 | 1 | 1 |
LCM = 2 × 2 × 3 × 5 × 5 = 2² × 3 × 5² = 300
LCM(12, 15, 75) = 300
Method 4: The GCF formula
LCM(a, b) = a × b ÷ GCF(a, b)
Find the greatest common divisor / GCF / HCF first, usually with the Euclidean algorithm, then divide the product by it. For three or more numbers, chain the formula: LCM of the first two, then LCM of that result and the next number.
- GCF(21, 14) = 7, so LCM = 21 × 14 ÷ 7 = 42
- GCF(42, 38) = 2, so LCM = 42 × 38 ÷ 2 = 798
LCM(21, 14, 38) = 798.
How to find the LCM: which method to use
| Situation | Best method | Why |
|---|---|---|
| Numbers up to 12 | Listing multiples | Fast, and shows what "common multiple" means |
| Two larger numbers | GCF formula | One division and one multiplication |
| Three or more numbers | Prime factorization or division method | Handles any count in one table |
| Exam working required | Division method | The layout examiners expect (class 5 to 10) |
Three or more numbers
All four methods extend to three or more numbers. The division table gains a column per number, and the GCF formula is applied pairwise. Worked examples: LCM of 3 or more numbers.
Common mistakes
- Lowest power instead of highest. 12 = 2² × 3 and 18 = 2 × 3². The LCM uses 2² and 3², not 2 and 3. Taking the lowest powers gives the GCF, 6.
- Multiplying everything. 12 × 18 = 216 is a common multiple, not the least one. Divide by the GCF, 6, to get 36.
- Stopping the ladder early. In the division method, keep dividing until every entry is 1.
- Mixing LCM and GCF. The GCF is never larger than the numbers; the LCM is never smaller than them.
Where the LCM is used in daily life
Adding fractions uses the LCM of the denominators. Two buses that leave every 12 and 18 minutes leave together again after 36 minutes. Two gears with 12 and 18 teeth return to their starting position after 36 teeth pass.
Frequently asked questions
How do I find the LCM of two numbers?
Use LCM(a, b) = a × b ÷ GCF(a, b). For 12 and 18 the GCF is 6, so LCM = 12 × 18 ÷ 6 = 36. For small numbers, listing multiples is just as quick.
How many methods are there to find the LCM?
Four standard methods: listing multiples, prime factorization, the division (ladder) method and the GCF formula. A Venn diagram is a fifth way to lay out prime factors. The LCM calculator shows all of them.
How do you find the LCM by hand?
Write each number as a product of primes, take every prime at its highest power, and multiply. 12 = 2² × 3 and 18 = 2 × 3², so LCM = 2² × 3² = 36. No calculator is needed.
How do you explain the LCM to a child?
Count in 4s and count in 6s. Write both lists. The first number that shows up in both lists, 12, is the LCM. Skip counting and a highlighted number make the idea visible.
What is the easiest way to find the LCM?
Check whether the larger number is a multiple of the smaller one. If yes, the larger number is the LCM. If no, use prime factorization, which works for any numbers of any size.
How do I find the LCM of 6 numbers?
Prime factorization: factor all six, take each prime at its highest power, multiply. LCM(2, 3, 4, 5, 6, 7) = 420. The calculator takes up to 15 numbers at once.
What are common mistakes when finding the LCM?
Three mistakes are common: taking the lowest power of a prime instead of the highest, multiplying all the numbers together, and confusing the LCM with the GCF. The GCF is always smaller than or equal to the numbers; the LCM is always larger than or equal to them.
Is listing multiples always a good method?
No. Listing multiples works for small numbers, up to about 12, and for teaching the idea. For larger numbers the lists get long, so prime factorization or the division method is faster.
Sources
- Wikipedia: Least common multiple: definition, notation, LCM × GCD = a × b
- Wolfram MathWorld: Least Common Multiple: formal properties
- Common Core 6.NS.B.4: GCF and LCM in grade 6