LCM Calculator
Find the least common multiple of 2 to 10 whole numbers, with the prime factorization, the GCF and every step shown.
Separate numbers with commas or spaces, for example 12, 18, 30.
180 is the smallest number that 12, 18 and 30 all divide into evenly.
How each number reaches the LCM
Multiply each number by its factor to reach the LCM. These factors build a common denominator.
The next common multiples are 360, 540 and 720. Every common multiple is a multiple of 180.
Your calculation
The same steps the calculator runs, with your numbers.
Prime factorization table
The LCM row takes the highest power in each column. The GCF row takes the lowest.
Multiples up to the LCM
The LCM is the first number that appears in every row.
| Number | Multiples |
|---|
Whole numbers from 1 to 10¹² are accepted. The LCM is computed exactly, even when it has more digits than a standard calculator shows.
What Is the Least Common Multiple?
A multiple of a number is that number times a whole number: the multiples of 4 are 4, 8, 12, 16 and so on. A common multiple of two or more numbers appears in all of their lists. The least common multiple (LCM), also called the lowest common multiple, is the smallest positive common multiple.
For 4 and 6, the multiples of 4 are 4, 8, 12, 16, 20, 24 and the multiples of 6 are 6, 12, 18, 24. The common multiples are 12, 24 and so on, so the LCM is 12. Every other common multiple is a multiple of the LCM.
The LCM of the denominators of two fractions is their least common denominator, which is why the LCM appears whenever fractions are added or compared. It also answers timing questions, such as when two repeating events next happen together.
How to Use the LCM Calculator
- Enter the numbers. Type 2 to 10 whole numbers separated by commas or spaces, such as 12, 18, 30. Each number can be as large as 1,000,000,000,000.
- Read the LCM. The large number is the least common multiple, with its prime factorization beside it.
- Use the details. The next card shows what to multiply each number by to reach the LCM, which gives the factors for a common denominator. Below it you get the GCF, the steps with your numbers, a prime factorization table and the lists of multiples.
Results update as you type. The quick examples cover two numbers, three numbers, numbers with no common factor and the LCM of 1 to 10.
LCM Formula with the GCF
The identity LCM × GCF = a × b holds for two numbers: 36 × 6 = 216 = 12 × 18. For three or more numbers, apply the formula two at a time.
GCF with Euclid's Algorithm
Euclid's algorithm replaces the larger number with the remainder until the remainder is 0. It finds the GCF of large numbers without factoring them, so the LCM follows quickly.
How to Find the LCM Manually
Method 1: List the multiples
Write multiples of each number until one appears in every list. For 4 and 6: 4, 8, 12 and 6, 12, so the LCM is 12. This works well for small numbers but becomes slow when the LCM is large.
Method 2: Prime factorization
Factor each number into primes, take the highest power of each prime that appears, and multiply. For 12, 18 and 30: 12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5. The highest powers are 2², 3² and 5, so the LCM is 4 × 9 × 5 = 180.
Method 3: The ladder (division) method
Divide by a prime that goes into at least two of the numbers, carry down any number it does not divide, and repeat until no prime divides two of the remaining numbers. Multiply all the divisors and the numbers left at the bottom.
LCM Table for Common Number Pairs
Each row lists the LCM, the GCF and a check that their product equals the product of the two numbers.
| Numbers | LCM | GCF | LCM × GCF |
|---|---|---|---|
| 2 and 3 | 6 | 1 | 6 |
| 3 and 4 | 12 | 1 | 12 |
| 4 and 6 | 12 | 2 | 24 |
| 6 and 8 | 24 | 2 | 48 |
| 8 and 12 | 24 | 4 | 96 |
| 9 and 12 | 36 | 3 | 108 |
| 10 and 15 | 30 | 5 | 150 |
| 12 and 15 | 60 | 3 | 180 |
| 12 and 18 | 36 | 6 | 216 |
| 15 and 20 | 60 | 5 | 300 |
| 16 and 24 | 48 | 8 | 384 |
| 24 and 36 | 72 | 12 | 864 |
Two well known results: the LCM of every number from 1 to 10 is 2,520, and the LCM of 1 to 20 is 232,792,560.
LCM vs GCF
| LCM | GCF | |
|---|---|---|
| Meaning | Smallest shared multiple | Largest shared divisor |
| Prime powers | Highest of each | Lowest of shared |
| Size | At least the largest number | At most the smallest number |
| 12 and 18 | 36 | 6 |
| Used for | Adding fractions, schedules | Simplifying fractions |
Understanding Your Result
The LCM is never smaller than the largest number. It equals the largest number when every other number divides it: the LCM of 4 and 12 is 12.
When no two numbers share a prime factor, the LCM is their product: LCM(5, 7) = 35. If any pair shares a factor, the LCM is smaller than the product: 8, 9 and 21 multiply to 1,512, but 9 and 21 share the factor 3, so the LCM is 504.
The factor next to each number in the second card is the multiplier for a common denominator.
Where the LCM Is Used
Adding and subtracting fractions
To add 1/4 + 1/6, use the LCM of 4 and 6, which is 12: 3/12 + 2/12 = 5/12. For 2/9 + 5/12, the LCM of 9 and 12 is 36, giving 8/36 + 15/36 = 23/36.
Schedules and repeating events
If one bus leaves every 12 minutes and another every 18 minutes, and both leave together at 8:00, they next leave together after LCM(12, 18) = 36 minutes, at 8:36.
Packaging and buying in sets
Hot dogs sold in packs of 10 and buns in packs of 8 match exactly at LCM(10, 8) = 40, which is 4 packs of hot dogs and 5 packs of buns.
Common Mistakes
- Mixing up LCM and GCF. For 12 and 18, 36 is the LCM and 6 is the GCF. The LCM is the bigger of the two.
- Multiplying the numbers together. 4 × 6 = 24 is a common multiple, but the least one is 12. The product is the LCM only when the numbers share no factor.
- Using the lowest prime powers. Lowest powers give the GCF. The LCM needs the highest power of every prime, including primes that appear in only one number.
- Extending the two number formula wrongly. 12 × 18 × 30 ÷ 6 = 1,080, not 180. Work two numbers at a time instead.
- Including 0. Every number divides 0, so the least positive common multiple is used and 0 is left out.
Related Calculations
The LCM is the first step in adding fractions with different denominators, which the Fraction Calculator does automatically. The Ratio Calculator uses the greatest common factor to simplify ratios such as 1920 : 1080 to 16 : 9.
Prime factors and powers also appear in the Exponent Calculator, and the Root Calculator uses the same prime factorization to simplify radicals such as √72 = 6√2.
Frequently Asked Questions
How do you find the LCM of two numbers?
The quickest method for two numbers uses the greatest common factor: LCM(a, b) = a × b ÷ GCF(a, b). For 12 and 18, the GCF is 6, so the LCM is 12 × 18 ÷ 6 = 216 ÷ 6 = 36. For small numbers you can also list multiples until one appears in both lists: 12, 24, 36 and 18, 36, so the LCM is 36. For large numbers, find the GCF with Euclid's algorithm first. For 252 and 105 the GCF is 21, so the LCM is 252 × 105 ÷ 21 = 1,260.
What is the LCM of 12 and 18?
The LCM of 12 and 18 is 36. In prime factors, 12 = 2² × 3 and 18 = 2 × 3². The LCM takes the highest power of each prime, which is 2² × 3² = 4 × 9 = 36. You can check it: 36 ÷ 12 = 3 and 36 ÷ 18 = 2, and no smaller positive number is divisible by both. The GCF of 12 and 18 is 6, and the product of the LCM and GCF equals the product of the numbers: 36 × 6 = 216 = 12 × 18.
What is the difference between LCM and GCF?
The least common multiple is the smallest number that every given number divides into, while the greatest common factor is the largest number that divides every given number. For 12 and 18 the LCM is 36 and the GCF is 6. In prime factors, the LCM uses the highest power of each prime and the GCF uses the lowest power of the primes they share. The LCM is never smaller than the largest number, and the GCF is never larger than the smallest number. Use the LCM to add fractions and line up repeating events, and the GCF to simplify fractions.
How do you find the LCM of three numbers?
Write each number as a product of primes, then multiply the highest power of every prime that appears. For 12, 18 and 30: 12 = 2² × 3, 18 = 2 × 3² and 30 = 2 × 3 × 5. The highest powers are 2², 3² and 5, so the LCM is 4 × 9 × 5 = 180. You can also work two numbers at a time: LCM(12, 18) = 36, then LCM(36, 30) = 180. Do not divide the product of all three by their common GCF, because 12 × 18 × 30 ÷ 6 = 1,080, which is a common multiple but not the least one.
Why do you need the LCM to add fractions?
Fractions can be added only when they have the same denominator, and the LCM of the denominators is the smallest one that works, called the least common denominator. For 1/4 + 1/6, the LCM of 4 and 6 is 12, so 1/4 = 3/12 and 1/6 = 2/12, and the sum is 5/12. For 2/9 + 5/12, the LCM of 9 and 12 is 36, giving 8/36 + 15/36 = 23/36. Any common multiple works, such as 24 for the first example, but the LCM keeps the numbers small and often avoids simplifying at the end.
Sources & References
- OpenStax. Prealgebra 2e, 2.5 Prime Factorization and the Least Common Multiple (listing and prime factor methods). openstax.org
- OpenStax. Prealgebra 2e, 4.5 Add and Subtract Fractions with Different Denominators (least common denominator). openstax.org
- OpenStax. Elementary Algebra 2e, 7.1 Greatest Common Factor and Factor by Grouping. openstax.org
- NIST Dictionary of Algorithms and Data Structures. Euclid's algorithm. xlinux.nist.gov
Results are for educational purposes. Every worked number on this page was recomputed with the same functions the calculator uses. Methodology reviewed: 2026.