Use the Language of Algebra

Find Factors, Prime Factorizations, and Least Common Multiples

The numbers 2, 4, 6, 8, 10, 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2.

Similarly, a multiple of 3 would be the product of a counting number and 3.

We could find the multiples of any number by continuing this process.

| Counting Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | {: valign=”top”}|———- | Multiples of 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | {: valign=”top”}| Multiples of 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | {: valign=”top”}| Multiples of 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 | {: valign=”top”}| Multiples of 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | {: valign=”top”}| Multiples of 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 | {: valign=”top”}| Multiples of 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 | {: valign=”top”}| Multiples of 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 | {: valign=”top”}| Multiples of 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 | {: valign=”top”}{: .unnumbered summary=”This table has 13 columns, 8 rows and a header row. The header row labels each column: counting number, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first column labels each row: multiples of 2, multiples of 3, multiples of 4, multiples of 5, multiples of 6, multiples of 7, multiples of 8, multiples of 9. The column labeled 1 has the following values: 2, 3, 4, 5, 6, 7, 8, 9. The column labeled 2 has the following values: 4, 6, 8, 10, 12, 14, 16, 18. The column labeled 3 has the following values: 6, 9, 12, 15, 18, 21, 24, 27. The column labeled 4 has the following values: 8, 12, 16, 20, 24, 28, 32, 36. The column labeled 5 has the following values: 10, 15, 20, 25, 30, 35, 40, 45. The column labeled 6 has the following values: 12, 18, 24, 30, 36, 42, 48, 54. The column labeled 7 has the following values: 14, 21, 28, 35, 42, 49, 56, 63. The column labeled 8 has the following values: 16, 24, 32, 40, 48, 56, 64, 72. The column labeled 9 has the following values: 18, 27, 36, 45, 54, 63, 72, 81. The column labeled 10 has the following values: 20, 30, 40, 50, 60, 70, 80, 90. The column labeled 11 has the following values: 22, 33, 44, 55, 66, 77, 88, 99. The column labeled 12 has the following values: 24, 36, 48, 60, 72, 84, 96, 108.”}

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, $15÷3$

is 5, so 15 is $5·3.$

If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since $8·9=72,$

we say that 8 and 9 are factors of 72. When we write $72=8·9,$

we say we have factored 72.

Other ways to factor 72 are $1·72,2·36,3·24,4·18,$

and $6·12.$

The number 72 has many factors: $1,2,3,4,6,8,9,12,18,24,36,$

and $72.$

Some numbers, such as 72, have many factors. Other numbers have only two factors. A prime number is a counting number greater than 1 whose only factors are 1 and itself.

The counting numbers from 2 to 20 are listed in the table with their factors. Make sure to agree with the “prime” or “composite” label for each!

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful in many topics in this course.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime. Otherwise it is easy to lose track of the prime numbers.

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

One of the reasons we look at primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 using their prime factors.

Notice that the prime factors of 12 $(2·2·3)$

and the prime factors of 18 $(2·3·3)$

are included in the LCM $(2·2·3·3).$

So 36 is the least common multiple of 12 and 18.

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

Use Variables and Algebraic Symbols

In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a variable and letters commonly used for variables are $x,y,a,b,c.$

A number whose value always remains the same is called a constant.

To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using. There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We’ll list the symbols used to indicate these operations below.

When two quantities have the same value, we say they are equal and connect them with an equal sign.

On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “<” and “>”.

The expressions $a<b$

or $a>b$

can be read from left to right or right to left, though in English we usually read from left to right. In general,

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols now.

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. A sentence has a subject and a verb. In algebra, we have expressions and equations.

Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.

An equation is two expressions linked by an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.

Suppose we need to multiply 2 nine times. We could write this as $2·2·2·2·2·2·2·2·2.$

This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write $2·2·2$

as $2^3$

and $2·2·2·2·2·2·2·2·2$

as $2^9.$

In expressions such as $2^3,$

the 2 is called the base and the 3 is called the exponent. The exponent tells us how many times we need to multiply the base.

<div data-type="note" markdown="1">

We say $2^3$

is in exponential notation and $2·2·2$

is in expanded notation.

means multiply a by itself, n times.

The expression $a^n$

is read a to the $n^{th}$

power.

</div>

While we read $a^n$

as $“a$

to the $n^{th}$

power”, we usually read:

We’ll see later why $a^2$

and $a^3$

have special names.

[link] shows how we read some expressions with exponents.

| Expression | In Words | | {: valign=”top”}|———- | 7² | 7 to the second power or | 7 squared | {: valign=”top”}| 5³ | 5 to the third power or | 5 cubed | {: valign=”top”}| 9⁴ | 9 to the fourth power | | {: valign=”top”}| 12⁵ | 12 to the fifth power | | {: valign=”top”}{: summary=”This table shows four expressions and words to describe these. The expressions described are 7 to the second power or 7 squared, 5 to the third power or 5 cubed, 9 to the fourth power and 12 to the fifth.”}

Simplify Expressions Using the Order of Operations

To simplify an expression means to do all the math possible. For example, to simplify $4·2+1$

we would first multiply $4·2$

to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression $4+3·7.$

Some students simplify this getting 49, by adding $4+3$

and then multiplying that result by 7. Others get 25, by multiplying $3·7$

first and then adding 4.

The same expression should give the same result. So mathematicians established some guidelines that are called the order of operations.

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase “Please Excuse My Dear Aunt Sally”.

It’s good that “My Dear” goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.

Similarly, “Aunt Sally” goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

Simplify: $5+2^3+3\left[6-3\left(4-2\right)\right].$

Are there any parentheses (or other grouping symbols)? Yes.
Focus on the parentheses that are inside the brackets. Subtract.
Continue inside the brackets and multiply.
Continue inside the brackets and subtract.
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes. Simplify exponents.
Is there any multiplication or division? Yes.
Multiply.
Is there any addition of subtraction? Yes.
Add.
Add.

Evaluate an Expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.

Identify and Combine Like Terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.

Examples of terms are $7,y,5x^2,9a,$

and $b^5.$

The constant that multiplies the variable is called the coefficient.

Think of the coefficient as the number in front of the variable. The coefficient of the term $3x$

is 3. When we write $x,$

the coefficient is 1, since $x=1·x.$

Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms.

Look at the following 6 terms. Which ones seem to have traits in common?

We say,

and $4$

are like terms.

and $3x$

are like terms.

and $9n^2$

are like terms.

If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

Translate an English Phrase to an Algebraic Expression

We listed many operation symbols that are used in algebra. Now, we will use them to translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. [link] summarizes them.

Look closely at these phrases using the four operations:

Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.

We look carefully at the words to help us distinguish between multiplying a sum and adding a product.

Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.

The expressions in the next example will be used in the typical coin mixture problems we will see soon.

Key Concepts

• Divisibility Tests

  ---

  A number is divisible by:

  ---

    2 if the last digit is 0, 2, 4, 6, or 8.

  ---

    3 if the sum of the digits is divisible by 3.

  ---

    5 if the last digit is 5 or 0.

  ---

    6 if it is divisible by both 2 and 3.

  ---

    10 if it ends with 0.

• How to find the prime factorization of a composite number.

  ---

  1. Find two factors whose product is the given number, and use these numbers to create two branches.
  2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
  3. If a factor is not prime, write it as the product of two factors and continue the process.
  4. Write the composite number as the product of all the circled primes.
• How To Find the least common multiple using the prime factors method.

  ---

  1. Write each number as a product of primes.
  2. List the primes of each number. Match primes vertically when possible.
  3. Bring down the columns.
  4. Multiply the factors.
• Equality Symbol

  ---

  $a=b$

  is read “a is equal to b.”

  ---

  The symbol “=” is called the equal sign.

• Inequality

  ---

  

• Inequality Symbols

  ---

  Inequality Symbols	Words
  {: valign=”top”}	———-
  $a\ne b$

  a is not equal to b.
  {: valign=”top”}	$a<b$

  a is less than b.
  {: valign=”top”}	$a\le b$

  a is less than or equal to b.
  {: valign=”top”}	$a>b$

  a is greater than b.
  {: valign=”top”}	$a\ge b$

  | a is greater than or equal to b. | {: valign=”top”}{: summary=”The table describes inequality symbols in words. The symbols described are a is not equal to b, a is less than b, a is less than or equal to b, a is greater then b, a is greater than or equal to b.”}

• Grouping Symbols

  ---

  $\begin{array}{cccccc}\text{Parentheses}\hfill & & & & & \left(\right)\hfill \\ \text{Brackets}\hfill & & & & & \left[\right]\hfill \\ \text{Braces}\hfill & & & & & \left\{\right\}\hfill \end{array}$
• Exponential Notation

  ---

  $a^n$

  means multiply a by itself, n times.

  ---

  The expression

  $a^n$

  is read a to the

  $n^{th}$

  power.

• Simplify an Expression

  ---

  To simplify an expression, do all operations in the expression.

• How to use the order of operations.

  ---

  1. Parentheses and Other Grouping Symbols
     • Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
  2. Exponents
     • Simplify all expressions with exponents.
  3. Multiplication and Division
     • Perform all multiplication and division in order from left to right. These operations have equal priority.
  4. Addition and Subtraction
     • Perform all addition and subtraction in order from left to right. These operations have equal priority.
• How to combine like terms.

  ---

  1. Identify like terms.
  2. Rearrange the expression so like terms are together.
  3. Add or subtract the coefficients and keep the same variable for each group of like terms.

  Operation	Phrase	Expression
  Addition	a plus b the sum of $a$ and b a increased by b b more than a the total of a and b b added to a	$a+b$
  Subtraction	a minus $b$ the difference of a and b a decreased by b b less than a b subtracted from a	$a-b$
  Multiplication	a times b the product of $a$ and $b$ twice a	$a·b,ab,a(b),(a)(b)$ $2a$
  Division	a divided by b the quotient of a and b the ratio of a and b b divided into a	$a÷b,a\text{/}b,\frac{a}{b},b\overline{)a}$

You can also download for free at http://cnx.org/contents/02776133-d49d-49cb-bfaa-67c7f61b25a1@4.13

Attribution:

• For questions regarding this license, please contact partners@openstaxcollege.org.
• If you use this textbook as a bibliographic reference, then you should cite it as follows: OpenStax College, Intermediate Algebra. OpenStax CNX. http://cnx.org/contents/02776133-d49d-49cb-bfaa-67c7f61b25a1@4.13.
• If you redistribute this textbook in a print format, then you must include on every physical page the following attribution: "Download for free at http://cnx.org/contents/02776133-d49d-49cb-bfaa-67c7f61b25a1@4.13."
• If you redistribute part of this textbook, then you must retain in every digital format page view (including but not limited to EPUB, PDF, and HTML) and on every physical printed page the following attribution: "Download for free at http://cnx.org/contents/02776133-d49d-49cb-bfaa-67c7f61b25a1@4.13."