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LCM Calculator (Least Common Multiple)

Find the least common multiple (LCM) of two to six whole numbers, worked out step by step from the GCF and by prime factorization.

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Least common multiple (LCM)

60

LCM(4, 6, 10)

Step by step

LCM(4, 6) = 4 × 6 ÷ GCF 2 = 12

LCM(12, 10) = 12 × 10 ÷ GCF 2 = 60

Using prime factorization

4 = 2²

6 = 2 × 3

10 = 2 × 5

LCM = each prime raised to its highest power: 2² × 3 × 5 = 60

LCM(a, b) = |a × b| ÷ GCF(a, b); with more numbers it's worked out in pairs. Negative numbers are taken as absolute values.

Keep analyzing

For estimates only. Double-check the numbers before making important decisions.

What the LCM is

The least common multiple of two or more numbers is the smallest positive whole number that all of them divide into evenly. The multiples of 4 are 4, 8, 12, 16… and the multiples of 6 are 6, 12, 18… so the LCM of 4 and 6 is 12.

How to find the LCM

LCM(a, b) = |a × b| ÷ GCF(a, b)
LCM(a, b, c) = LCM(LCM(a, b), c)
By prime factorization: every prime factor, raised to its highest power

The GCF method is the fastest for large numbers. The prime factorization method shows why the answer works: 4 = 2², 6 = 2 × 3 and 10 = 2 × 5, so the LCM is 2² × 3 × 5 = 60.

Zeros and negative numbers

  • If any of the numbers is 0, the LCM is 0 by convention.
  • Negative numbers are treated as their absolute values.
  • With large inputs the LCM can get huge, so it's calculated with exact integer arithmetic.

Common uses

  • Adding and subtracting fractions: the LCM of the denominators is the least common denominator. 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
  • Repeating events: if one bus comes every 12 minutes and another every 18, they arrive together every 36 minutes.
  • Matching package sizes: hot dogs come in packs of 10 and buns in packs of 8. The LCM is 40, so buy 4 packs of hot dogs and 5 packs of buns to have none left over.

Example

The LCM of 4, 6 and 10 is 60. The calculator works in pairs: first LCM(4, 6) = 12, then LCM(12, 10) = 60.

  1. LCM(4, 6) = 4 × 6 ÷ GCF 2 = 24 ÷ 2 = 12
  2. LCM(12, 10) = 12 × 10 ÷ GCF 2 = 120 ÷ 2 = 60
  3. Check with primes: 2² × 3 × 5 = 60

Common mistakes

  • Multiplying all the numbers together: that gives a common multiple, but usually not the least one (4 × 6 × 10 = 240, not 60).
  • Taking the common prime factors with their lowest powers: that gives the GCF, not the LCM.

Frequently asked questions

How are the LCM and the GCF related?

For two numbers, LCM × GCF = |a × b|. For 12 and 18, the GCF is 6 and the LCM is 36, and 6 × 36 = 216 = 12 × 18.

What is the LCM used for in fractions?

It's the least common denominator. It lets you add or subtract fractions with different denominators while keeping the numbers as small as possible.

What is the LCM of two prime numbers?

Their product, because they share no factors other than 1. The LCM of 7 and 11 is 77.

Can the LCM be smaller than the numbers?

No. For positive numbers, the LCM is always at least as large as the biggest number. It equals that number when the others divide into it, as in LCM(3, 12) = 12.

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