Algebra/Chapter 1/Factoring and Divisibility

1.4: Factoring and Divisibility

By now, I hope you are familiar with the addition, subtraction, multiplication, and exponentiation of whole numbers. Division of whole numbers, however, works a little differently from the other three operators. This is because you can not always divide integers evenly.

You might've noticed that some numbers divide evenly, while others do not. For example:

• ${\displaystyle 12\div 4=3}$, because ${\displaystyle 4+4+4=12}$, and there are 3 fours that are needed to add to 12.
• ${\displaystyle 12\div 5}$ doesn't divide evenly because ${\displaystyle 5+5=10}$, which is too little, while ${\displaystyle 5+5+5=15}$, which is too much.

A factor is a number that can divide evenly into another number. From the example above, 4 is a factor of 12, but 5 is not a factor of 12.

We can try to find other factors of 12. With some trial and error, we can find all of the factors of 12, which are 1, 2, 3, 4, 6, and 12. These numbers all divide evenly into 12, while other numbers, such as 7 or 13, don't divide evenly into 12 and hence are not factors of 12.

Keep in mind that we can look at other numbers besides 12, such as 6, which have factors of 1, 2, 3, and 6. The number 7 only has two factors though, which are 1 and 7 itself.

On the other hand, a multiple is a number that can be divided evenly by another number. Using the example above again, 12 is a multiple of 4, but 12 is not a multiple of 5. The term "multiple" comes from the fact that 4 can be multiplied by something (in this case, 3) to get 12, while 5 cannot be multiplied by something to get 12.

Like factors, we can find the multiples of a particular number, let's say 4, and find all of the multiples of it. However, instead of guessing and checking, we can see that all multiples of 4 has to be 4 multiplied by something, regardless of that something. So, all of the multiples of 4 will look like ${\displaystyle 4\times 1}$, ${\displaystyle 4\times 2}$, ${\displaystyle 4\times 3}$, and so on, which yields ${\displaystyle 4,8,12,16,20,24,28,\ldots }$

Unlike factors, the list of multiples of a particular number goes on forever instead of stopping.