LCM Calculator

How to use the LCM calculator

Enter two or more positive whole numbers separated by commas or spaces. You get the least common multiple, the greatest common factor, the multiplier that takes each number to the LCM (exactly what you need when building a common denominator) and the next few common multiples. The steps show the prime-factor method with a table, the GCF shortcut, and — for small numbers — the lists of multiples.

What is the least common multiple?

A multiple of a number is the result of multiplying it by a whole number: the multiples of 4 are 4, 8, 12, 16, 20 and so on. A common multiple of several numbers is a number that appears in all of their lists, and the least common multiple is the smallest positive one. It is also called the lowest common multiple, and when the numbers are denominators it is the least common denominator (LCD).

Methods and formulas

Prime factorization

Factor each number into primes. The LCM is the product of every prime that appears, each raised to the highest power found in any of the numbers.

Compare this with the GCF, which takes only the shared primes with the lowest powers.

The GCF shortcut

For 4 and 6, the GCF is 2, so the LCM is 4 × 6 ÷ 2 = 12. Then LCM(12, 10): the GCF is 2, so 12 × 10 ÷ 2 = 60. This is the fastest method for large numbers because the GCF can be found quickly with the Euclidean algorithm.

Listing multiples

For small numbers, simply list multiples until you find the first one they share. Multiples of 10: 10, 20, 30, 40, 50, 60. Check: 60 ÷ 4 = 15 and 60 ÷ 6 = 10, both whole, so 60 is the LCM.

Worked example: adding fractions

To add 1/4 + 1/6 + 3/10, you need a common denominator. The LCM of 4, 6 and 10 is 60, and the calculator's "LCM ÷ each number" row (15, 10, 6) gives the multipliers: 1/4 = 15/60, 1/6 = 10/60 and 3/10 = 18/60. Adding gives 43/60. Using the LCM keeps the numbers as small as possible; multiplying the denominators together would give 240 and a fraction that still needs simplifying. The fraction calculator applies this automatically.

Real-world uses of the LCM

• Scheduling: if one bus leaves every 4 minutes, another every 6 and a third every 10, they all depart together every 60 minutes.
• Buying in packs: hot dogs come in packs of 10 and buns in packs of 8. The smallest number of each that matches exactly is LCM(10, 8) = 40: four packs of hot dogs and five packs of buns.
• Gears and cycles: two meshing gears with 12 and 18 teeth return to their starting alignment after LCM(12, 18) = 36 teeth have passed.
• Music and rhythm: a 3-beat and a 4-beat pattern line up again every 12 beats.

Useful facts

• For two numbers, GCF × LCM = the product of the numbers.
• If two numbers are coprime (GCF = 1), their LCM is simply their product: LCM(8, 9) = 72.
• If one number divides another, the LCM is the larger number: LCM(6, 18) = 18.
• The LCM is always at least as large as the largest number in the set.