Common Multiples — Definition & Examples

Common Multiples

Multiples of a number are numbers we get as a result of multiplying an integer by another integer.

Common multiples are multiples that two numbers have in common. For example, what are some common multiples of $3$ and $5$? We can start by listing some multiples of $3$: $$3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33…$$ Next, list the multiples of $5$: $$5, 10, 15, 20, 25, 30, 35, 40…$$ Now, the numbers common to both lists are: $15, 30$. The smallest positive integer on the list is the least common multiple. This is sometimes abbreviated as "LCM."

$15$ and $30$ are common multiples of $3$ and $5$.

Any two numbers have an infinite amount of common multiples.

A trick to find common multiples for two numbers is to multiply the two numbers together and find the multiples of that number.

Let’s take a look at an example. Find the common multiples of $2$ and $3$.

First, multiply the two numbers together. $$2 \cdot 3 = 6$$ Next, find the multiples of $6$. $$6, 12, 18, 24 …$$ These are the common multiples of $2$ and $3$. We can check this by listing the multiples of $2$ and $3$. $$\text{Multiples of }2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24…$$ $$\text{Multiples of }3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 ….$$ The numbers common to both lists are: $6, 12, 18, 24$, which are identical to the common multiples we found above using the trick.

We are now going to take a look at some of the types of problems you might encounter about common multiples.

$2, 7$: $2 \cdot 7$ = $14$, and $36$ is not a multiple of $14$.

$3, 14$: $3 \cdot 14$ = $42$, and $36$ is not a multiple of $42$.

$2, 18$: $2 \cdot 18$ = $36$, so $36$ is a common multiple of these two numbers.

$6, 7$: $6 \cdot 7$ = $42$, and $36$ is not a multiple of $42$.

We can use the trick we learned previously to solve for the common multiples.

First, multiply the two numbers together: $$3 \cdot 5 = 15$$ Next, find the multiples of $15$. $$15, 30, 45 …$$ From the choices above, $15$ is the only number listed, so $15$ is the common multiple of $3$ and $5$.

Common Multiples

To understand what a common multiple is, you must first understand what a multiple is. As a refresher, a multiple of an integer is the product of that integer and another integer. For example, some multiples of $10$ are $\cdots , -30, -20, -10, 0, 10, 20, 30, \cdots$

Common multiples are just multiples that two (or more) numbers share, and they can be negative or positive! For example, $-8$ is a common multiple of $2$ and $4$ because $-8$ is a multiple of both $2$ and $4.$

Let's do another example to practice!

Solution

Common multiples of $2, 5,$ and $15$ are multiples that these numbers share, and they can be negative or positive! Let's go through the list and check which answer choices are multiples of $2, 5,$ and $15$:

• $15$ can't be a common multiple of $2, 5,$ and $15$ because $15$ isn't a multiple of $2$
• $10$ can't be a common multiple of $2, 5,$ and $15$ because $10$ isn't a multiple of $15$
• $30$ is a common multiple of $2, 5,$ and $15$ because $30$ is a multiple of $2,$ a multiple of $5,$ and a multiple of $15.$
• $20$ can't be a common multiple of $2, 5,$ and $15$ because $20$ isn't a multiple of $15$
• $-60$ is a common multiple of $2, 5,$ and $15$ because $-60$ is a multiple of $2,$ a multiple of $5,$ and a multiple of $15.$
• $150$ is a common multiple of $2, 5,$ and $15$ because $150$ is a multiple of $2,$ a multiple of $5,$ and a multiple of $15.$

(Video) How to Find Common Multiples

by IMA Videos

Here's a video by IMA Videos that explains what common multiples are and how to find them.

$\large{\text{Summary}}$

The common multiples of two numbers are the multiples that appear for both numbers, similar to how common factors are the factors that appear in both numbers.

Suppose we want to find the common factors of $3$ and $5$. We can start by listing out their factors. $$\begin{array}{c|cccc} 3:&3&6&9&12&15&18&21&24&27&30\\ 5:&5&10&15&20&25&30&35 \end{array}$$ The common multiples are the multiples they both share. $$\begin{array}{c|cccc} 3:&3&6&9&12&\color{#29A603}{15}&18&21&24&27&\color{#8925AE}{30}\\ 5:&5&10&\color{#29A603}{15}&20&25&\color{#8925AE}{30}&35 \end{array}$$

Once you find the the smallest common multiple, any multiple of that will be a common multiple of the two. This smallest number is also called the lowest common multiple. If we are asked to find this number, the problem might just say "find the LCM." LCM is just an abbreviation.

Take our example, the smallest common multiple was $15$. The multiples of $15$ will be common multiples of $3$ and $5$. $$\begin{array}{c|cccc} 15:&15&30&45&60 \end{array}$$

All of the numbers in this list have $3$, $5$, and $15$ as common divisors.