How to find the least common multiple (LCM)

The least common multiple is the first positive number that two or more whole numbers divide into evenly. For small numbers, we can find it by listing multiples. When the numbers get larger, prime factors give us a quicker way to find the first match.

Learn Pre-Algebra with meStep-by-step videos, lesson notes, and practice for whole numbers, fractions, decimals, and more.

Explore Pre-Algebra →

List positive multiples until you find a match

To find the LCM of $3$ and $4$, start with the positive multiples of $4$. The first three are $4$, $8$, and $12$. The first of those that $3$ divides evenly is $12$.

The first match is 12, so it is the least common multiple.

Use prime factors for larger numbers

Write $20$ and $75$ as products of primes. To make a multiple of both, include each prime to the highest power it has in either factorization.

$\begin{array}{l}20=2^2·5\\ 75=3·5^2\\ \mathrm{LCM}=2^2·3·5^2=300\end{array}$

We need two factors of $2$ from $20$, and one factor of $3$ and two of $5$ from $75$. Those are the smallest prime powers that meet both requirements.

Dividing $300$ by $20$ gives $15$, and dividing by $75$ gives $4$. That checks divisibility. The prime powers explain why no smaller positive common multiple works.

Use the same rule with more than two numbers

For $10$, $15$, and $24$, the highest powers are $2^3$, $3$, and $5$. Multiply them to get $120$, which all three numbers divide evenly.

Full Pre-Algebra course

Learn Pre-Algebra with me

We’ll work through numbers, fractions, decimals, and other Pre-Algebra topics, with the steps and examples explained as we go.

Explore my Pre-Algebra course →

• Video lessons
• Lesson notes
• Practice problems

Included with membership, along with every other Krista King Math course.