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$$\textbf{Multiples: A Visual Approach}$$

$$\textbf{Breaking Down the Example}$$

Let's look at the above example in a bit more depth. It discusses counting by $5$s in a way that is sometimes called skip counting.

A multiple of $5$ is the product of $5$ and any integer. Here is a partial list of multiples of $5$:

$$\begin{align} 5 \cdot 1 &= \color{29a603}5, \text{ so $5$ is a multiple of $5$}\\ 5 \cdot 2 &= \color{29a603}10, \text{ so $10$ is a multiple of $5$}\\ 5 \cdot 3 &= \color{29a603}15, \text{ so $15$ is a multiple of $5$}\\ 5 \cdot 4 &= \color{29a603}20, \text{ so $20$ is a multiple of $5$}\\ 5 \cdot 5 &= \color{29a603}25, \text{ so $25$ is a multiple of $5$}\\ \end{align}$$

These are just a few of them. In reality the list goes on forever. Remember, multiples of $5$ are products of $5$ and any integer. There are infinitely many integers. That means there are infinite multiples of $5.$ For example: $$\begin{align} 5 \cdot -6 &= \color{29a603}-30, \text{ so $-30$ is a multiple of $5$}\\ 5 \cdot 0 &= \color{29a603}0, \text{ so $0$ is a multiple of $5$}\\ 5 \cdot 20 &= \color{29a603}100, \text{ so $100$ is a multiple of $5$}\\ \end{align}$$

In particular, multiplying $5$ by a negative integer gives a multiple of $5$. But these resulting numbers are negative. They lie to the left of $0$ on the number line. Finally, we can multiply $5$ by $0$, which is an integer, to get $0$.

A multiple of $12$ is a product of $12$ with some integer. $$\begin{align} 12 \cdot -1 &= -12, \text{ so $-12$ is a multiple of $12$}\\ 12 \cdot -2 &= -24, \text{ so $-24$ is a multiple of $12$}\\ 12 \cdot 4 &= 48, \text{ so $48$ is a multiple of $12$}\\ 12 \cdot 1 &= 12, \text{ so $12$ is a multiple of $12$}\\ 12 \cdot 0 &= 0, \text{ so $0$ is a multiple of $12$}\\ \end{align}$$

$12 \cdot \dfrac{1}{2}=6.$ Since $\dfrac{1}{2}$ is not an integer, $6$ cannot be a multiple of $12.$ Similarly, $12 \cdot \dfrac{1}{12}=1.$ Since $\dfrac{1}{12}$ is not an integer, $1$ cannot be a multiple of $12.$

Notice that a number is always a multiple of itself, because the product of any number and the integer $1$ is that number. In addition, $0$ is a multiple of any number, because the product of any number and the integer $0$ is $0.$

$$\textbf{Defining Multiples}$$

Throughout this explanation we are going to focus on the natural numbers. They are the main numbers that are really relevant for this topic. Remember the natural numbers are the counting numbers: $1, 2, 3, 4, \dots$

A multiple of any natural number is a number which can be evenly divided by that number. That is, no remainder is produced from this procedure.

Example: The number $33$ can be divided by $3$ without any remainder. So $33$ is a multiple of $3$ ($3 \cdot 11 = 33$). If you'd like, this can be checked with the division fact

$$33 \div 3 = 11$$

The multiples of a number are formed by multiplying it with any natural number, such as $1,2,3,\cdots.$ We could also multiply by a negative integer. A number can have infinitely many of these. On the other hand, this resulting number has the original number as a divisor.

Examples

Example: $5, 10,15,20,25,30, \cdots$ are multiples of $5$. They are written as follows:

$$\begin{align} \color{29a603}5 &= 5 \times 1 \\ \color{29a603}10 &= 5 \times 2 \\ \color{29a603}15 &= 5 \times 3 \\ \color{29a603}20 &= 5 \times 4 \\ \color{29a603}25 &= 5 \times 5 \\ \color{29a603}30 &= 5 \times 6 \\ & \cdots \end{align}$$

Example: $7, 14, 21, 28, 35, 42, \cdots$ are multiples of $7$. They are written as follows:

$$\begin{align} \color{29a603}7 &= 7 \times 1 \\ \color{29a603}14 &= 7 \times 2 \\ \color{29a603}21 &= 7 \times 3 \\ \color{29a603}28 &= 7 \times 4 \\ \color{29a603}35 &= 7 \times 5 \\ \color{29a603}42 &= 7 \times 6 \\ & \cdots \end{align}$$

Some Important Facts:

• Every number is a multiple of $1$.
• Zero is a multiple of every number. This is because zero times any number is zero. However, the number itself is usually referred to as the "first" number in the list. This is because you multiply the number by $1$ to get the number itself.
• The first positive multiple of every number is the number itself. This is because $1$ is the multiplicative identity.
• There are infinitely many multiples of any number. They will get bigger and bigger. If you place them all on a number line, they'll be evenly spaced apart.
• Squaring a number produces a multiple of that number.
• For any two positive integers, there is a least common multiple of those two numbers.

(Video) What is a Multiple?

by Benjamin Cogswell

This is a video by Benjamin Cogswell that covers our topic of the day.

$\large{\text{Summary}}$

Multiples are numbers that follow a rule of repeated addition. In other words, if you add a number to itself, you get a multiple. Each time you keep adding itself, you get a new multiple. If your original number was positive, the numbers you list out will get bigger. If your original number was negative, the numbers you list out will get smaller. This is very different from a factor. Factors are numbers you multiply together to get the original number.

Let's look at the multiples of $2$. There are too many to list them all, but we can list the first couple. $$\begin{array}{c|l} 2&2\\ 4&2+2\\ 6&2+2+2\\ 8&2+2+2+2\\ 10&2+2+2+2+2 \end{array}$$ The smallest multiple of $2$ is the number itself, $2$. However, the list goes on forever without stopping.

This process is very different from asking "find the common multiples." That question is asking you to compare two lists like the one above.

Before moving on, let's do another example that uses some bigger numbers. We're going to try our hand at listing multiples of $11$. We will use the same process as the previous example.

$$\begin{array}{c|l} 11&11\\ 22&11+11\\ 33&11+11+11\\ 44&11+11+11+11\\ 55&11+11+11+11+11 \end{array}$$