All skills Math

Learn Index · Number Sense and Numeration Grade 6–8

Least common multiple

The smallest number that is a multiple of both numbers.

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What is the LCM?

The least common multiple (LCM) of two numbers is the smallest positive number that both divide into.

For 4 and 6:

  • Multiples of 4: 4, 8, 12, 16, 20, 24 …
  • Multiples of 6: 6, 12, 18, 24, 30 …
  • Common multiples: 12, 24, 36 …
  • LCM = 12

Method 1: List multiples

Write multiples of each number until one appears in both lists. This works well for small numbers. Tip: list multiples of the bigger number and check each against the smaller one.

LCM of 8 and 12: multiples of 12 are 12, 24 … Is 12 a multiple of 8? No. Is 24? Yes. LCM = 24.

Method 2: Prime factorization

  1. Write each number’s prime factorization.
  2. For every prime that appears, take the highest power.
  3. Multiply them.

Example: LCM of 18 and 24

  • 18 = 2 × 3²
  • 24 = 2³ × 3
  • Highest powers: 2³ and 3²
  • LCM = 8 × 9 = 72

Method 3: Use the GCF

For two numbers: LCM = (a × b) ÷ GCF(a, b)

Example: 18 and 24. GCF = 6. LCM = (18 × 24) ÷ 6 = 432 ÷ 6 = 72.

Special cases

Situation LCM Example
One number divides the other The bigger number LCM(5, 20) = 20
The numbers are relatively prime Their product LCM(4, 9) = 36

Using the LCM

Adding fractions: The least common denominator is the LCM of the denominators.

  • 5/6 + 3/8 → LCM(6, 8) = 24 → 20/24 + 9/24 = 29/24

When events line up: A bus comes every 12 minutes and a train every 18 minutes. They arrive together at 8:00. When will they next arrive together? LCM(12, 18) = 36 → 8:36.

Packages: Hot dogs in packs of 10, buns in packs of 8. Smallest number of each to match: LCM = 40 (4 packs of hot dogs, 5 packs of buns).

Worked examples

Example 1: LCM of 4 and 6.

12.

Example 2: LCM of 15 and 20.

15 = 3 × 5; 20 = 2² × 5 → 2² × 3 × 5 = 60.

Example 3: LCM of 3, 4 and 10.

Primes: 2² (from 4), 3, 5 → 60.

Common mistakes

  • Multiplying the two numbers without checking for a smaller common multiple. 4 × 6 = 24, but the LCM is 12.
  • Mixing up GCF and LCM. The GCF is no bigger than the smaller number; the LCM is at least the bigger number.
  • Taking the lowest power of each prime instead of the highest.

Try it yourself

  1. LCM of 6 and 9
  2. LCM of 7 and 8
  3. LCM of 12 and 30
  4. Find the least common denominator for 1/4 + 5/6.
  5. Two lights blink every 6 s and 8 s. They blink together now. When next?

Answers: 1) 18 2) 56 3) 60 4) 12 5) in 24 seconds

Quick recap

  • The least common multiple is the smallest shared multiple.
  • Find it by listing, prime factorization (highest powers) or LCM = ab ÷ GCF.
  • It gives the least common denominator for fractions.