Use this free LCM calculator to find the least common multiple, also called the lowest common multiple, of two to ten whole numbers. It shows the working four ways: listing the multiples, prime factorization, the GCF formula and the ladder method. It also gives the GCF, the LCM as primes, the multiplier for each number and the next common multiples. The Common Multiples tab lists the first 5, 10 or 20 common multiples of your numbers. Tables list the LCM of the most common pairs, the other names for the LCM and which method to use. It is free and instant, and your numbers stay in your browser.
What is the least common multiple?
The least common multiple (LCM) of two or more whole numbers is the smallest number that all of them divide into exactly. The multiples of 4 are 4, 8, 12, 16, 20, 24 and so on, the multiples of 6 are 6, 12, 18, 24, 30, and the multiples of 8 are 8, 16, 24, 32. The first number on all three lists is 24, so the LCM of 4, 6 and 8 is 24. It is also called the lowest common multiple, the smallest common multiple or the minimum common multiple. They are all the same number.
Method 1: list the multiples
Write the multiples of each number in order and stop at the first one they all share. For 4 and 6, the lists meet at 12. It is easy to see and good for small numbers, but it takes a long time when the LCM is big, so the calculator lists the multiples only when no number needs more than 40 of them.
Method 2: prime factorization
Break each number into primes: 4 = 2², 6 = 2 × 3 and 8 = 2³. Then take each prime the most times it appears in any of them: 2 three times (from 8) and 3 once (from 6). So the LCM is 2³ × 3 = 24. The same primes, taken the fewest times instead, give the greatest common factor.
Method 3: the GCF formula
For two numbers, LCM(a, b) = a × b ÷ GCF(a, b). For 12 and 18 the GCF is 6, so the LCM is 12 × 18 ÷ 6 = 36. For more numbers, repeat it: LCM(4, 6) = 12, and then LCM(12, 8) = 12 × 8 ÷ 4 = 24. It needs no factoring, so it is the best way to get the LCM of big numbers. The GCF Calculator finds the GCF with Euclid's algorithm.
Method 4: the ladder method
Write the numbers in a row and divide by a prime that goes into at least two of them, carrying down the numbers it does not divide. Repeat until no prime goes into two numbers. For 4, 6 and 8: divide by 2 to get 2, 3, 4, then by 2 again to get 1, 3, 2. Multiply the divisors and what is left: 2 × 2 × 1 × 3 × 2 = 24. This is a favorite in schools for three or more numbers.
LCM and GCF
For two numbers, GCF × LCM = the product of the numbers. For 12 and 18, 6 × 36 = 216 = 12 × 18. The GCF is the biggest number that goes into both, and the LCM is the smallest number both go into, so the two answers are a pair. The calculator shows the GCF next to the LCM.
When the LCM is the product
If two numbers share no factor except 1, they are coprime, and their LCM is just their product. 7 and 8 are coprime, so their LCM is 56. If one number divides the other, the LCM is the bigger one: the LCM of 5 and 10 is 10.
Common multiples
A common multiple is any number that all of them divide into. Every common multiple is a multiple of the LCM, so the common multiples of 4 and 6 are 12, 24, 36, 48, 60 and on forever. The Common Multiples tab lists the first 5, 10 or 20 of them.
LCM and fractions
To add fractions with different denominators you need a common denominator, and the best one is the lowest common denominator (LCD), which is the LCM of the denominators. For 1/4 + 1/6 the LCD is 12, so the sum is 3/12 + 2/12 = 5/12. The Fraction Calculator does this for you.
Zero and negative numbers
Zero is a multiple of every number, so the LCM of any list that includes 0 is 0. For negative numbers the calculator uses the absolute value, so the LCM of −4 and 6 is 12.
A real use: when two things line up again
If one bus leaves every 4 minutes and another every 6 minutes, and they leave together now, they will leave together again after the LCM of 4 and 6, which is 12 minutes. The same sum works for gears, for repeating lights and for planning tasks that repeat on different schedules.
