Least Common Multiple Formula

The Formula

\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}

When to use: The first number that appears in both times tables—where two counting sequences land on the same value.

Quick Example

LCM of 4 and 6: Multiples of 4 (4,8,12,16...) and 6 (6,12,18...). LCM = 12.

Notation

\text{LCM}(a, b) or \text{lcm}(a, b) denotes the least common multiple of a and b

What This Formula Means

The smallest positive integer that is divisible by each of two or more given numbers—where their multiples first coincide.

The first number that appears in both times tables—where two counting sequences land on the same value.

Formal View

\text{lcm}(a, b) = \min\{m \in \mathbb{Z}^+ : a \mid m \text{ and } b \mid m\}. Via prime factorization: \text{lcm}(a,b) = \prod p_i^{\max(\alpha_i, \beta_i)}. Relation: \gcd(a,b) \cdot \text{lcm}(a,b) = |ab|.

Worked Examples

Example 1

easy
Find the LCM of 12 and 18.

Solution

  1. 1
    Prime-factor each number: 12 = 2^2 \times 3 and 18 = 2 \times 3^2.
  2. 2
    For the LCM, keep the highest exponent of each prime that appears: 2^2 and 3^2.
  3. 3
    Multiply those factors: 2^2 \times 3^2 = 4 \times 9 = 36, so the LCM is 36.

Answer

36
The LCM is the product of all prime factors, each raised to the highest power from either number. The LCM is the smallest number that both original numbers divide into evenly.

Example 2

medium
Find the LCM of 8, 12, and 15.

Common Mistakes

  • Multiplying the two numbers to find LCM — LCM of 4 and 6 is 12, not 4 \times 6 = 24 (the product only works when the numbers share no common factors)
  • Taking the smaller power of each prime instead of the larger — for 4 = 2^2 and 6 = 2 \times 3, LCM uses 2^2 and 3^1, giving 12, not 2^1 = 2
  • Confusing LCM with GCF — LCM of 4 and 6 is 12 (smallest shared multiple), while GCF is 2 (largest shared factor)

Why This Formula Matters

Essential for adding fractions with different denominators: \frac{1}{4} + \frac{1}{6} uses \text{LCM}(4,6) = 12.

Frequently Asked Questions

What is the Least Common Multiple formula?

The smallest positive integer that is divisible by each of two or more given numbers—where their multiples first coincide.

How do you use the Least Common Multiple formula?

The first number that appears in both times tables—where two counting sequences land on the same value.

What do the symbols mean in the Least Common Multiple formula?

\text{LCM}(a, b) or \text{lcm}(a, b) denotes the least common multiple of a and b

Why is the Least Common Multiple formula important in Math?

Essential for adding fractions with different denominators: \frac{1}{4} + \frac{1}{6} uses \text{LCM}(4,6) = 12.

What do students get wrong about Least Common Multiple?

Using prime factorization: LCM uses the larger power of each prime.

What should I learn before the Least Common Multiple formula?

Before studying the Least Common Multiple formula, you should understand: multiples.

← Back to Least Common Multiple See All Examples Practice Problems

Related Concepts

MultiplesAdding Fractions