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 2 and a counting number.

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.

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 15 by 3, 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.

How to Find the Prime Factorization of a Composite Number

Factor 48.

Solution

We say $2·2·2·2·3$ is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer.

If we first factored 48 in a different way, for example as $6·8,$ the result would still be the same. Finish the prime factorization and verify this for yourself.

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.

How to Find the Least Common Multiple Using the Prime Factors Method

Find the least common multiple (LCM) of 12 and 18 using the prime factors method.

Solution

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.

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.

Table 8 shows how we read some expressions with exponents.

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.

Simplify: $18÷6+4\left(5-2\right).$

Solution

Parentheses? Yes, subtract first.
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right.
Any other multiplication or division? Yes.
Multiply.
Any other multiplication of division? No.
Any addition or subtraction? Yes.
Add.

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

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

Solution

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.

Evaluate when $x=4:$ ⓐ $x^2$ ⓑ $3^x$ ⓒ $2x^2+3x+8.$

Solution

ⓐ

Use definition of exponent.
Simplify.

ⓑ

Use definition of exponent.
Simplify.

ⓒ

Follow the order of operations.

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,

$7$ and $4$ are like terms.

$5x$ and $3x$ are like terms.

$n^2$ 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.

How To Combine Like Terms

Simplify: $2x^2+3x+7+x^2+4x+5.$

Solution

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. Table 14 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.

Translate each English phrase into an algebraic expression:

ⓐ the difference of $14x$ and 9 ⓑ the quotient of $8y^2$ and 3 ⓒ twelve more than $y$ ⓓ seven less than $49x^2$

Solution

ⓐ The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.

ⓑ The key word is quotient, which tells us the operation is division.

ⓒ The key words are more than. They tell us the operation is addition. More than means “added to.”

$\begin{array}{c}\hfill \text{twelve more than}y\hfill \\ \hfill \text{twelve added to}y\hfill \\ \hfill y+12\hfill \end{array}$

ⓓ The key words are less than. They tell us to subtract. Less than means “subtracted from.”

$\begin{array}{c}\hfill \text{seven less than}49x^2\hfill \\ \hfill \text{seven subtracted from}49x^2\hfill \\ \hfill 49x^2-7\hfill \end{array}$

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

Translate the English phrase into an algebraic expression:

ⓐ eight times the sum of x and y ⓑ the sum of eight times x and y

Solution

There are two operation words—times tells us to multiply and sum tells us to add.

ⓐ Because we are multiplying 8 times the sum, we need parentheses around the sum of x and y, $\left(x+y\right).$ This forces us to determine the sum first. (Remember the order of operations.)

$\begin{array}{c}\hfill \text{eight times the sum of}x\text{and}y\hfill \\ \hfill 8(x+y)\hfill \end{array}$

ⓑ To take a sum, we look for the words of and and to see what is being added. Here we are taking the sum of eight times x and y.

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.

Simplify Expressions with Absolute Value

A negative number is a number less than 0. The negative numbers are to the left of zero on the number line. See Figure 1.

You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers $2$ and $\mathrm{-2}$ are the same distance from zero, each one is called the opposite of the other. The opposite of $2$ is $\mathrm{-2},$ and the opposite of $\mathrm{-2}$ is $2.$

Figure 2 illustrates the definition.

We saw that numbers such as 3 and $\mathrm{-3}$ are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

For example,

Figure 3 illustrates this idea.

The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.

In the next example, we’ll order expressions with absolute values.

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.

In the next example, we simplify the expressions inside absolute value bars first just as we do with parentheses.

Add and Subtract Integers

So far, we have only used the counting numbers and the whole numbers.

Our work with opposites gives us a way to define the integers. The whole numbers and their opposites are called the integers. The integers are the numbers $\dots \mathrm{-3},\mathrm{-2},\mathrm{-1},0,1,2,3\text{…}$

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one color (blue) represent positive. The other color (red) will represent the negatives.

If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

We will use the counters to show how to add:

The first example, $5+3,$ adds 5 positives and 3 positives—both positives.

The second example, $\mathrm{-5}+\left(\mathrm{-3}\right),$ adds 5 negatives and 3 negatives—both negatives.

When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8—either 8 positives or 8 negatives.

So what happens when the signs are different? Let’s add $\mathrm{-5}+3$ and $5+\left(\mathrm{-3}\right).$

When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.

Add: ⓐ $\mathrm{-1}+\left(\mathrm{-4}\right)$ ⓑ $\mathrm{-1}+5$ ⓒ $1+\left(\mathrm{-5}\right).$

Solution

ⓐ

1 negative plus 4 negatives is 5 negatives

ⓑ

There are more positives, so the sum is positive.

ⓒ

There are more negatives, so the sum is negative.

We will continue to use counters to model the subtraction. Perhaps when you were younger, you read $“5-3”$ as “5 take away 3.” When you use counters, you can think of subtraction the same way!

We will use the counters to show to subtract:

The first example, $5-3,$ we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, $\mathrm{-5}-\left(\mathrm{-3}\right),$ we subtract 3 negatives from 5 negatives and end up with 2 negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. If we don’t have the number of counters needed to take away, we add neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.

Let’s look at $\mathrm{-5}-3$ and $5-\left(\mathrm{-3}\right).$

Model the first number.
We now add the needed neutral pairs.
We remove the number of counters modeled by the second number.
Count what is left.

Subtract: ⓐ $3-1$ ⓑ $\mathrm{-3}-\left(\mathrm{-1}\right)$ ⓒ $\mathrm{-3}-1$ ⓓ $3-\left(\mathrm{-1}\right).$

Solution

ⓐ

	$$
Take 1 negative from 3 negatives and get 2 negatives positives.

ⓑ

Take 1 positive from 3 negatives and get 2 negatives.

ⓒ

	$$
Take 1 positive from the one added neutral pair.

ⓓ

	$$
Take 1 negative from the one added neutral pair.

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In the last example, $\mathrm{-3}-1$ is the same as $\mathrm{-3}+\left(\mathrm{-1}\right)$ and $3-\left(\mathrm{-1}\right)$ is the same as $3+1.$ You will often see this idea, the Subtraction Property, written as follows:

What happens when there are more than three integers? We just use the order of operations as usual.

Multiply and Divide Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we are using the model just to help us discover the pattern.

We remember that $a·b$ means add a, b times.

The next two examples are more interesting. What does it mean to multiply 5 by $\mathrm{-3}?$ It means subtract $5,3$ times. Looking at subtraction as “taking away”, it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.

In summary:

Notice that for multiplication of two signed numbers, when the

What about division? Division is the inverse operation of multiplication. So, $15÷3=5$ because $5·3=15.$ In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.

Division follows the same rules as multiplication with regard to signs.

When we multiply a number by 1, the result is the same number. Each time we multiply a number by $\mathrm{-1},$ we get its opposite!

Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember Please Excuse My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

The last example showed us the difference between ${\left(\mathrm{-2}\right)}^4$ and $\text{−}{2}^4.$ This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

Evaluate $4x^2-2xy+3y^2$ when $x=2,y=\mathrm{-1}.$

Solution

	$$
Simplify exponents.
Multiply.
Subtract.
Add.

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

How to Solve Application Problems Using Integers

In the morning, the temperature in Kendallville, Indiana was 11 degrees. By mid-afternoon, the temperature had dropped to $\mathrm{-9}$ degrees. What was the difference in the morning and afternoon temperatures?

Solution