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Factors vs. Multiples

Multiples of a number are products of that number and an integer. They are usually written out in a list. Often we only ask for positive multiples. These are the multiples of a number which are also positive. We'll show an example of this later.

Factors of a number are integers that evenly divide into that number. For positive integers, each factor is less than or equal to the original number.

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

Before you start these problems, make sure to go over the basics of factors vs. multiples.

A number is a multiple of $5$ if it is the product of $5$ and an integer.

The positive multiples of $5$ are: $5, 10, 15, 20, 25, 30, 35 ...$

The correct answers are: $5, 15, 20, 25$

Factors of a number are integers that divide into that number without a remainder.

Positive factors of $10$ are: $1, 2, 5, 10$

Only the number itself (and the negative version of that number) can be both a factor and multiple.

Thus, $20$ and $-20$ are both factors and multiples of $20$.

Any number that satisfies this condition must be both a positive factor of $28$ and a positive multiple of $2$. The positive factors of $28$ are: $$1, 2, 4, 7, 14, 28$$ and the positive multiples of $2$ are: $$2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 ...$$ The numbers that are in both lists are $2, 4, 14$ and $28$. We conclude $2, 4, 14, 28$ are both positive factors of $28$ and positive multiples of $2$.

All whole numbers have factors and multiples. These terms are often confused because their definitions are very similar to one another. Factors and multiples are also closely related to each other. $\\$

Multiples

Multiples of a number are the numbers we get as a result of multiplying a number $x$ by an integer $y$. The product, $z$, is a multiple of both $x$ and $y$. For example, let $x=2$ and $y=3$. Then $z = xy = 2 \times 3 = 6$.

Thus, $6$ is a multiple of $2$. We can multiply $2$ by an integer $(3)$ to get $6$. With the same logic, $6$ is also a multiple of $3$. We can multiply $3$ by an integer $(2)$ to get $6$. In fact, a number will be a multiple of all of its factors. For instance, here is a list of factors of $12$:

$$1, 2, 3, 4, 6, 12$$

Then $12$ is a multiple for each number in the above list. Larger numbers usually have a greater number of factors. This excludes prime numbers, which only have two factors.

There's another way to think of multiples of numbers. It is thinking about counting by that number. This is also called skip-counting. For example, if we count by two, $2,4,6,8,10,12, \dots$, all of the numbers listed are multiples of two because they are all a product of $2$ and another number. i.e.

$$\{2\times{\color{#29A603}{1}},2\times{\color{#29A603}{2}},2\times{\color{#29A603}{3}},2\times{\color{#29A603}{4}},2\times{\color{#29A603}{5}},2\times{\color{#29A603}{6}}\}=\{2,4,6,8,10,12\}$$

Just like we can count by twos forever, we can count multiples of numbers forever. Each number has an infinite amount of multiples.

Now that we know what multiples are, let's move onto factors. Let's begin with the idea that every integer is actually a product of two other integers. The two integers that are multiplied together to create another integer are called factors. Each number, even prime numbers, are a product of two numbers. In the case of prime numbers, their factors are $1$ and itself. i.e. $11$ is the product of $11$ and $1$. Thus, $11$ and $1$ are factors of $11$. This goes along with the multiplication facts

$$11 \times 1 = 11$$ $$1 \times 11 = 11$$

Let's now consider composite numbers. Many composite numbers, like $24$, are the result of several different products.

$$(1\times 24=24; \space 2\times 12=24; \space 3\times 8=24; \space 4\times 6=24)$$

Each of those numbers are factors of $24$ because they can each be multiplied by another number to reach the target number, $24$.

In the above question we only asked for one pair of factors for $15$. However, there is one other pair we didn't list: $1$ and $15$.

TLDR: Given the numbers $x$, $y$, and $z$. Let $x\times y=z$. Then $x$ and $y$ are factors of $z$. Also, $z$ is a multiple of both $x$ and $y$.

(Video) Factors Vs. Multiples

by Mr. Voland

Here's a good video by Mr. Voland that explains the difference between factors and multiples. The two concepts are closely related. However, they don't exactly mean the same thing!

$\large{\text{Summary}}$

There are a finite number of factors, but an infinite amount of multiples.

He shows some helpful techniques using pictures to demonstrate. For example, if a number was a tree, its factors would be the roots. The multiples would be the leaves. This analogy might help you remember the difference.

You can think of your original number as the trunk of the tree. It connects the leaves and the roots.

Factors are smaller and finite. They're the numbers that make up the your number. So, for the tree example above, the trunk is $10$. The roots (factors) of the tree are $1, 2, 5,$ and $10$. There are only $4$ factors of $10$, which means that the number of factors are finite. You can physically count how many factors there are. They will all be positive integers less than the number on the trunk.

Multiples are larger and infinite. They're the numbers that can be made with your number. So, for the tree example above, the multiples (leaves) of $10$ are $20, 30, 40, 50, 60, 70, 80, 90,$ and $100$. Now, there are many more multiples of $10$. In fact, there are an infinite number of multiples of $10$! This means you cannot physically count all the multiples of $10$. You could start now, and by the time you're $85$ years old, you still won't be done counting!

Both factors and multiples have to do with multiplication. Let's take a look at a few examples and see if we can identify the difference between factors and multiples. If we list some factors and some multiples, maybe we can solve the puzzle!

First, notice $4,5$ are factors of $20$. These aren't all the factors. We just need a few for the moment.

Moreover, $20$ is a multiple of both $4$ and $5$.

What are the other factors of $20$?

$1,2,10,20$ are also factors of $20$. Notice that the factors are all whole numbers. This is true regardless of what number we're factoring.

What are some other multiples of $5$?$\\$ A number is a multiple of $5$ if we can multiply $5$ with a whole number and get that number.

In fact, for any number, there is an infinite amount of multiples! That is, it would take forever to list them all.

Some facts about factors and multiples

Factors $\\$ The factors of a number are smaller or equal to that number. $\\$ The factors of a number are integers. $\\$ For any number, there is a limited amount of factors.

Multiples $\\$ A multiple of a number is bigger or equal to that number. $\\$ A multiple of a number is an integer. $\\$ For any number, there is an infinite amount of multiples.

Try practicing with these concepts now.