LESSON: Least Common Multiple

READ: Find the LCM using Prime Factorization

Find the Least Common Multiple of Given Numbers Using Prime Factorization

Remember back to factoring numbers? We worked on using factor trees to factor numbers or to break down numbers into their primes. Take a look at this one.

$& \ \ \ \quad 12\\ & \ \quad \ \big / \ \ \big\backslash\\ & \quad \ 4 \quad \ \ 3\\ & \quad \big / \ \big\backslash\\ & \ \ 2 \ \ \ 2\\ & 2^2 \times 3$

We used a factor tree in this example to factor twelve down to the prime factors of 2 squared times 3.

We can also use prime factorization when looking for the least common multiple.

How can we use prime factorization to find the LCM?

If we wanted to find the LCM of two numbers without listing out all of the multiples, we could do it by using prime factorization.

Example

What is the LCM of 9 and 12?

First, we factor both numbers to their primes.

$& \quad \ \ 9 && \quad \ 12\\ & \quad \big / \ \ \big\backslash && \quad \big / \ \ \big\backslash\\ & \ 3 \qquad 3 && \ 3 \qquad 4\\ &&& \ \qquad \big / \ \big\backslash\\ &&& \qquad 2 \quad \ 2$

Next, we identify any shared primes. With 9 and 12, 3 is a shared prime number.

Then, we take the shared prime and multiply it with all of the other prime factors.

3 $\times$ 3 $\times$ 2 $\times$ 2

The red 3 is the shared prime factor.

The blue numbers are the other prime factors.

Our answer is 36. The LCM of 9 and 12 is 36.