There are often several ways to talk about the same idea. So far, we’ve seen that if [latex]m[/latex] is a multiple of [latex]n[/latex], we can say that [latex]m[/latex] is divisible by [latex]n[/latex]. We know that [latex]72[/latex] is the product of [latex]8[/latex] and [latex]9[/latex], so we can say [latex]72[/latex] is a multiple of [latex]8[/latex] and [latex]72[/latex] is a multiple of [latex]9[/latex]. We can also say [latex]72[/latex] is divisible by [latex]8[/latex] and by [latex]9[/latex]. Another way to talk about this is to say that [latex]8[/latex] and [latex]9[/latex] are factors of [latex]72[/latex]. When we write [latex]72=8\cdot 9[/latex] we can say that we have factored [latex]72[/latex].

In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.

For example, suppose a choreographer is planning a dance for a ballet recital. There are [latex]24[/latex] dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.

In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of [latex]24[/latex]. The table below summarizes the different ways that the choreographer can arrange the dancers.

What patterns do you see in the table above? Did you notice that the number of groups times the number of dancers per group is always [latex]24?[/latex] This makes sense, since there are always [latex]24[/latex] dancers.

You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of [latex]24[/latex], which are:

[latex]1,2,3,4,6,8,12,24[/latex]

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with [latex]1[/latex]. If the quotient is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.

In the following video we show how to find all the factors of [latex]30[/latex].

Identify Prime and Composite Numbers

Some numbers, like [latex]72[/latex], have many factors. Other numbers, such as [latex]7[/latex], have only two factors: [latex]1[/latex] and the number. A number with only two factors is called a prime number. A number with more than two factors is called a composite number. The number [latex]1[/latex] is neither prime nor composite. It has only one factor, itself.

The table below lists the counting numbers from [latex]2[/latex] through [latex]20[/latex] along with their factors. The highlighted numbers are prime, since each has only two factors.

Factors of the counting numbers from [latex]2[/latex] through [latex]20[/latex], with prime numbers highlighted

The prime numbers less than [latex]20[/latex] are [latex]2,3,5,7,11,13,17,\text{and }19[/latex]. There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than [latex]1[/latex] and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.

In the following video we show more examples of how to determine whether a number is prime or composite.

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