Common Multiple Calculator
Enter two to six whole numbers to find their least common multiple (LCM) and the first ten common multiples they share. The step-by-step panel shows the prime factorization method and the listing-multiples method so you can follow along. Results update as you type.
What is a common multiple?
A multiple of a number is what you get when you multiply that number by a positive whole number. The multiples of 4 are 4, 8, 12, 16, 20, 24... and so on forever. A common multiple of two or more numbers is a value that appears in every one of their multiple lists. 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 values that appear in both lists are 12, 24, 36, 48... Those are the common multiples of 4 and 6.
What is the least common multiple (LCM)?
The least common multiple is simply the smallest common multiple of a set of numbers. For 4 and 6 the LCM is 12, because 12 is the first value that appears in the multiple list of both numbers. Every other common multiple is a whole-number multiple of the LCM, so once you know the LCM you can generate the full list by multiplying it by 1, 2, 3, 4... The LCM is also called the least common denominator (LCD) when you are adding fractions, because it is the smallest denominator that both fractions can be rewritten with.
How to find the LCM using prime factorization
The most reliable method for larger numbers is prime factorization. Break each number down into its prime factors. For example, 12 = 2^2 x 3 and 18 = 2 x 3^2. Then take the highest power of every prime that appears: 2^2 x 3^2 = 4 x 9 = 36. So the LCM of 12 and 18 is 36. This method extends naturally to three or more numbers: collect all the prime factors across every number and always keep the highest exponent of each. The step-by-step panel above shows this working for your exact inputs.
The relationship between LCM and GCF
For any two positive integers a and b, the product of the LCM and the Greatest Common Factor (GCF, also called GCD) always equals the product of the two numbers: LCM(a, b) x GCF(a, b) = a x b. This identity is useful for a quick sanity check. If the GCF of 4 and 6 is 2, then LCM x 2 = 4 x 6 = 24, so LCM = 12, which matches. When you know the GCF already, you can find the LCM in one step using this formula. For three or more numbers the identity does not hold in the same simple form, but the prime factorization method still gives the correct answer.
Where common multiples are used in everyday maths
Common multiples appear in several practical situations. Adding fractions with unlike denominators requires finding the LCD, which is the LCM of the denominators, for example 1/4 + 1/6 becomes 3/12 + 2/12 = 5/12. Scheduling problems use the LCM to find when two repeating events coincide: if bus A comes every 8 minutes and bus B every 12 minutes, they meet at the LCM, which is 24 minutes, then again at 48, 72... Pattern and tile layouts use the LCM to find the shortest repeating unit. Understanding the LCM also underpins modular arithmetic and cryptography.
Common multiples of small number pairs
| Numbers | LCM | First 4 common multiples |
|---|---|---|
| 2 and 3 | 6 | 6, 12, 18, 24 |
| 2 and 4 | 4 | 4, 8, 12, 16 |
| 3 and 4 | 12 | 12, 24, 36, 48 |
| 4 and 6 | 12 | 12, 24, 36, 48 |
| 4 and 8 | 8 | 8, 16, 24, 32 |
| 5 and 10 | 10 | 10, 20, 30, 40 |
| 6 and 9 | 18 | 18, 36, 54, 72 |
| 8 and 12 | 24 | 24, 48, 72, 96 |
| 3 and 5 | 15 | 15, 30, 45, 60 |
| 7 and 11 | 77 | 77, 154, 231, 308 |
First four common multiples for frequently asked number pairs.
Frequently asked questions
What are the common multiples of 8 and 12?
The multiples of 8 are 8, 16, 24, 32, 40, 48... and the multiples of 12 are 12, 24, 36, 48, 60... The values that appear in both lists are 24, 48, 72, 96... The LCM is 24, and every subsequent common multiple is a further multiple of 24.
How is the LCM different from the GCF?
The LCM (least common multiple) is the smallest number that all your inputs divide into evenly. The GCF (greatest common factor) is the largest number that divides into all your inputs evenly. The LCM is always at least as large as the biggest number you entered, while the GCF is always no larger than the smallest.
How do you find the LCM of three numbers?
Use prime factorization: factor each number into primes, then for every prime that appears in any of the three factorizations take the highest exponent, and multiply those together. Alternatively, compute the LCM of the first two numbers, then compute the LCM of that result with the third number. The approach chains identically for four, five, or six numbers.
What is the least common denominator and how does it relate to the LCM?
The least common denominator (LCD) is the LCM of the denominators of a set of fractions. To add 1/4 and 1/6, find LCM(4, 6) = 12, then rewrite both fractions with denominator 12: 3/12 + 2/12 = 5/12. This calculator gives you the LCD directly when you enter the denominators as your numbers.
Are there infinitely many common multiples?
Yes. Because every common multiple is a whole-number multiple of the LCM, and the whole numbers go on forever, there are always infinitely many common multiples. This calculator lists the first several ones, but the sequence never ends.
Can two numbers share a common multiple equal to their product?
Yes, and this happens exactly when the two numbers are coprime (their GCF is 1). For example, 5 and 7 have a GCF of 1, so their LCM is 5 x 7 = 35. When numbers share factors larger than 1, their LCM is smaller than their product.