# The Language of Algebra

## Find Multiples and Factors


### Identify Multiples of Numbers

Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos:  She has  shoes in her closet.
The numbers  are called multiples of  Multiples of  can be written as the product of a counting number and  The first six multiples of  are given below.
A multiple of a number is the product of the number and a counting number. So a multiple of  would be the product of a counting number and  Below are the first six multiples of 
We can find the multiples of any number by continuing this process.  shows the multiples of  through  for the first twelve counting numbers.
Recognizing the patterns for multiples of  will be helpful to you as you continue in this course.
 shows the counting numbers from  to  Multiples of  are highlighted. Do you notice a pattern?
The last digit of each highlighted number in  is either  This is true for the product of  and any counting number. So, to tell if any number is a multiple of  look at the last digit. If it is  then the number is a multiple of 
Now let’s look at multiples of   highlights all of the multiples of  between  and  What do you notice about the multiples of 
All multiples of  end with either  or  Just like we identify multiples of  by looking at the last digit, we can identify multiples of  by looking at the last digit.
 highlights the multiples of  between  and  All multiples of  all end with a zero.
 highlights multiples of  The pattern for multiples of  is not as obvious as the patterns for multiples of 
Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of  is based on the sum of the digits. If the sum of the digits of a number is a multiple of  then the number itself is a multiple of  See .
Consider the number  The digits are  and  and their sum is  Since  is a multiple of  we know that  is also a multiple of 
Look back at the charts where you highlighted the multiples of  of  and of  Notice that the multiples of  are the numbers that are multiples of both  and  That is because  Likewise, since  the multiples of  are the numbers that are multiples of both  and 

### Use Common Divisibility Tests

Another way to say that  is a multiple of  is to say that  is divisible by  In fact,  is  so  is  Notice in  that  is not a multiple  When we divided  by  we did not get a counting number, so  is not divisible by 
Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests.  summarizes divisibility tests for some of the counting numbers between one and ten.

### Find All the Factors of a Number

There are often several ways to talk about the same idea. So
far, we’ve seen that if  is a multiple of  we can say that  is divisible by  We know that  is the product of  and  so we can say  is a multiple of  and  is a multiple of  We can also say  is divisible by  and by  Another way to talk about this is to say that  and  are factors of  When we write  we can say that we have factored 
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  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   summarizes the different ways that the choreographer can arrange the dancers.
What patterns do you see in ? Did you notice that the number of groups times the number of dancers per group is always  This makes sense, since there are always  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  which are:
We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with  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.

### Identify Prime and Composite Numbers

Some numbers, like  have many factors. Other numbers, such as  have only two factors:  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  is neither prime nor composite. It has only one factor, itself.
 lists the counting numbers from  through  along with their factors. The highlighted numbers are prime, since each has only two factors.
The prime numbers less than  are  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  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.

### Key Concepts

1. Factors   If , then   and   are factors of , and  is the product of   and .
2. Find all the factors of a counting number.

3. Determine if a number is prime.


### Practice Makes Perfect

Identify Multiples of Numbers 
In the following exercises, list all the multiples less than  for the given number.
Use Common Divisibility Tests 
In the following exercises, use the divisibility tests to determine whether each number is divisible by 
Find All the Factors of a Number 
In the following exercises, find all the factors of the given number.
Identify Prime and Composite Numbers 
In the following exercises, determine if the given number is prime or composite.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?