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1 Foundational Skills

Topics Covered[1]

In case you missed something in class, or just want to review a specific topic covered in this Module, here is a list of topics covered:

  1. Use Variables and Algebraic Symbols
  2. Use Negatives and Opposites
  3. Multiply Integers
  4. Divide Integers
  5. Identify Multiples
  6. Use Common Divisibility Tests
  7. Find all the Factors of the Given Number
  8. Key Concepts

Use Variables and Algebraic Symbol[2]

Greg and Alex have the same birthday, but they were born in different years. This year Greg is 20 years old and Alex is 23, so Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The 3 years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age g. Then we could use g + 3 to represent Alex’s age. See below.

Greg’s age Alex’s age
12 15
20 23
35 38
g g + 3

Letters are used to represent variables. Letters often used for variables are x, y, a, b, and c.

Variable

A variable is a letter that represents a number or quantity whose value may change.

Constant

A constant is a number whose value always stays the same.

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. The symbols for the four basic arithmetic operations are: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

Operation Notation Say: The result is…
Addition a + b a plus b the sum of a and b
Subtraction a − b a minus b the difference of a and b
Multiplication a · b, (a)(b), (a)b, a(b) a times b The product of a and b
Division \displaystyle a \div b, a/b, \frac{a}{b}, \frac{b}{a}
 a divided by b The quotient of a and b

In algebra, the cross symbol, “ד, is not used to show multiplication because that symbol may cause confusion. Does 3xy mean 3×y (three times y) or 3·x·y (three times x times y)? To make it clear, use “•” or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

  • The sum of 5 and 3 means add 5 plus 3, which we write as 5 + 3.
  • The difference of 9 and 2 means subtract 9 minus 2, which we write as 9 − 2.
  • The product of 4 and 8 means multiply 4 times 8, which we can write as 4 · 8.
  • The quotient of 20 and 5 means divide 20 by 5, which we can write as 20 ÷ 5.

Try it! – Translating Algebraic Expressions

Translate from algebra to words:

a. 12 + 14

b. (30)(5)

c. 64 ÷ 8

d. x − y

Solution A (click to reveal)

Problem Number Translations
a. 12 + 14
12 plus 14
the sum of twelve and fourteen

 

Solution B (click to reveal)

Problem Number Translations
b.
 (30)(5)
30 times 5
the product of thirty and five

 

 

Solution C (click to reveal)

Problem Number Translations
c. 64 ÷ 8
64 divided by 8
the quotient of sixty-four and eight

 

Solution D (click to reveal)

Problem Number Translations
d. x − y
x minus y
the difference of x and y

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

a = b is read: a is equal to b

The symbol = is called the equal sign.

An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So, if we know that b is greater than a, it means that b is to the right of a on the number line. We use the symbols “<” and “>” for inequalities.

a < b is read: a is less than b

a is to the left of b on the number line

The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.

a > b is read: a is greater than b

a is to the right of b on the number line

The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.

The expressions a < b and a > b can be read from left-to-right or right-to-left, though we usually read from left-to-right in English. In general:

a < b is equivalent to b > a.    For example, 7 < 11 is equivalent to 11 > 7.
a > b is equivalent to b < a.    For example, 17 > 4 is equivalent to 4 < 17.

When we write an inequality symbol with a line under it, such as a ≤ b, it means a < b or a = b. We read this as a is less than or equal to b. Also, if we put a slash through an equal sign, ≠, it means not equal.

We summarize the symbols of equality and inequality below.

Algebraic Notation Say
a = b a is equal to b
a ≠ b a is not equal to b
a < b a is less than b
a > b a is greater than b
a ≤ b a is less than or equal to b
a ≥ b a is greater than or equal to b

The symbols “<“ and “>” each have a smaller side and a larger side.

smaller side < larger side

larger side > smaller side

The smaller side of the symbol faces the smaller number and the larger faces the larger number.

Try it! – Translating Algebraic Inequality Expressions

Translate from algebra to word:

a. 20 ≤ 35

b. 11 ≠ (15 − 3)

c. 9 > (10 ÷ 2)

d. (x + 2) < 10

Solution A (click to reveal)

Algebraic Word
a. 20 \leq 35 20 is less than or equal to 35

 

Solution B (click to reveal)

Algebraic Word
b.  11 \leq (15−3) 11 is not equal to 15 minus 3

 

Solution C (click to reveal)

Algebraic Word
c. 9 > (10 \div 2) 9 is greater than 10 divided by 2

 

Solution D (click to reveal)

Algebraic Word
d. (x + 2) < 10 x plus 2 is less than 10

Try it! – Using Inequality Symbols in Expressions and Equations 

The information in the table compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol =, <, or > in each expression to compare the fuel economy of the cars.

This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled “Car” and the second “Fuel economy (mpg)”. To the right of the ‘Car’ row are the labels: “Prius”, “Mini Cooper”, “Toyota Corolla”, “Versa”, “Honda Fit”. Each of these columns contains an image of the labeled car model. To the right of the “Fuel economy (mpg)” row are the algebraic equations: the letter p, the equals symbol, the number forty-eight; the letter m, the equals symbol, the number twenty-seven; the letter c, the equals symbol, the number twenty-eight; the letter v, the equals symbol, the number twenty-six; and the letter f, the equals symbol, the number twenty-seven.
(credit: modification of work by Bernard Goldbach, Wikimedia Commons)

a. MPG of Prius_____ MPG of Mini Cooper

b. MPG of Versa_____ MPG of Fit

c. MPG of Mini Cooper_____ MPG of Fit

d. MPG of Corolla_____ MPG of Versa

e. MPG of Corolla_____ MPG of Prius

Solution A (click to reveal)

Steps Algebraic
Initial Problem MPG of Prius__MPG of Mini Cooper
Find the values in the chart. 48__27
Compare. 48 > 27
MPG of Prius > MPG of Mini Cooper

 

Solution B (click to reveal)

Steps Algebraic
Initial Problem MPG of Versa____MPG of Fit
Find the values in the chart. 26____27
Compare. 26 < 27
MPG of Versa < MPG of Fit

 

Solution C (click to reveal)

c.

Steps Algebraic
Initial Problem MPG of Mini Cooper____MPG of Fit
Find the values in the chart. 27____27
Compare. 27 = 27
MPG of Mini Cooper = MPG of Fit

 

Solution D (click to reveal)

d.

Steps Algebraic
Initial Problem MPG of Corolla____MPG of Versa
Find the values in the chart. 28____26
Compare. 28 > 26
MPG of Corolla > MPG of Versa

 

Solution E (click to reveal)

e.

Steps Algebraic
Initial Problem MPG of Corolla____MPG of Prius
Find the values in the chart. 28____48
Compare. 28 < 48
MPG of Corolla < MPG of Prius

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The Table below lists three of the most commonly used grouping symbols in algebra.

Common Grouping Symbols
parentheses (  )
brackets [  ]
braces {  }

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

    \begin{align*}\text{Ex. 1}&\quad 8(14-8)\\ \text{Ex. 2}&\quad 21 - 3\left[2 + 4\left(9 - 8\right)\right] \\ \text{Ex. 3}&\quad 24 \div \left[13 - 2\left[1\left(6 − 5\right) + 4\right]\right]\end{align*}

Use Negatives and Opposites[3]

Our work so far has only included counting numbers and whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than 0. The negative numbers are to the left of zero on the number line. See below.

 

A number line extends from negative 4 to 4. A bracket is under the values “negative 4” to “0” and is labeled “Negative numbers”. Another bracket is under the values 0 to 4 and labeled “positive numbers”. There is an arrow in between both brackets pointing upward to zero.
The number line shows the location of positive and negative numbers.

The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.

Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.

Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See the following number line.

 

A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled “larger”. An arrow below the number line extends from 1 towards negative 4 and is labeled “smaller”.
The numbers on a number line increase in value going from left to right and decrease in value going from right to left.

Remember that we use the notation:

a < b (read “a is less than b”) when a is to the left of b on the number line.

a > b (read “a is greater than b”) when a is to the right of b on the number line.

Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in the example below are called the integers. The integers are the numbers … -4, −3, −2, −1, 0, 1, 2, 3, 4…

 

A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.
All the marked numbers are called integers.

Try it! – Numbers, Inequalities, and Number Lines 

Order each of the following pairs of numbers using < or >:

a. 14___6

b. −1___9

c. −1___−4

d. 2___−20.

Solution (click to reveal)

It may be helpful to refer to the number line shown.

A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.

Reason Solution
a. 14 is to the right of 6 on the number line. 14___6
14 > 6
b. −1 is to the left of 9 on the number line. −1___9
−1 < 9
c. −1 is to the right of −4 on the number line. −1___−4
−1 > −4
d. 2 is to the right of −20 on the number line. 2___−20
2 > −20

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 −2 are the same distance from zero, they are called opposites. The opposite of 2 is −2, and the opposite of −2 is 2.

Opposite

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero. The following illustrates the definition:

A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.
The opposite of 3 is −3.

−a means the opposite of the number a.

The notation −a is read as “the opposite of a.”

Sometimes in algebra the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “−” used in three different ways.

Expression
 Explanation
10 − 4 Between two numbers, it indicates the operation of subtraction.
We read 10−4 as “10 minus 4.”
−8 In front of a number, it indicates a negative number.
We read −8 as “negative eight.”
−x In front of a variable, it indicates the opposite. We read −x as “the opposite of x.”
−(−2) Here there are two “−” signs. The one in the parentheses tells us the number is negative 2. The one outside the parentheses tells us to take the opposite of −2.   We read −(−2) as “the opposite of negative two.”

Try it! – Opposites on a Number Line

Find:

a. the opposite of 7

b. the opposite of −10

c. −(−6).

Solution (click to reveal)

Explanation
 Number Line
a. −7 is the same distance from 0 as 7, but on the opposite side of 0. .
The opposite of 7 is −7.
b. 10 is the same distance from 0 as −10, but on the opposite side of 0. .
The opposite of −10 is 10.
c. −(−6) .
The opposite of −(−6) is −6.

 

 Our work with opposites gives us a way to define the integers.

Integers

The whole numbers and their opposites are called the integers.

The integers are the numbers …−3, −2, −1, 0, 1, 2, 3…

When evaluating the opposite of a variable, we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether −x is positive or negative. We can see this in the example below.

Try it! – Evaluating Algebraic Expressions 

Evaluate

a. −x, when x = 8

b. −x, when x = −8.

Solution A (click to reveal)

a.

Steps
 Evaluation
To evaluate when x = 8 means to substitute 8 for x. −x
Substitute 8 for -x. -(8)
Write the opposite of 8. -8

 

Solution B (click to reveal)

b.

Steps Evaluation
To evaluate when x = -8 means to substitute –8 for x. −x
Substitute –8 for -x. – (-8)
Write the opposite of −8. 8

Multiply Integers[4]

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 will use the model just to help us discover the pattern.

We remember that a·b means add a, b times. Here, we are using the model just to help us discover the pattern.


Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”The next two examples are more interesting.  What does it mean to multiply 5 by −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. Then we take away 5 three times.


This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.In summary:

Sign Arithmetic Examples
+ \times + = + 5 \times 3 = 15
- \times + = - -5 \times 3 = -15
+ \times - = - 5 \times -3 = -15
- \times - = + -5 \times -3 = 15

Notice that for multiplication of two signed numbers, when the:

  • signs are the same, the product is positive.
  • signs are different, the product is negative.

We’ll put this all together in the chart below.

For the multiplication of two signed numbers:

Same signs Product Example
Positive · Positive Positive 7 · 4 = 28
Negative · Negative Positive −8 (−6) = 48
Different signs Product Example
Positive · Negative Negative 7 (−9) = −63
Negative · positive Negative −5 · 10 = −50

 

Try it! – Multiplying Numbers

Multiply:

a. −9 ·  3

b. −2 (−5)

c. 4 (−8)

d. 7 · 6

Solution (click to reveal)

Explanation Algebraic
a. Multiply, noting that the signs are different, so the product is negative. −9 · 3 = −27
b. Multiply, noting that the signs are the same, so the product is positive. −2(−5) = 10
c. Multiply with different signs. 4(−8) = −32
d. Multiply with the same signs. 7 · 6 = 42

When we multiply a number by 1, the result is the same number. What happens when we multiply a number by −1? Let’s multiply a positive number and then a negative number by −1 to see what we get.

Example 1 Example 2
−1 · 4 −1 (−3)
−4 3
−4 is the opposite of 4. 3 is the opposite of −3.

Each time we multiply a number by −1, we get its opposite!

−1 · a = −a

Multiplying a number by −1 gives its opposite.

Try it! – Multiplying Numbers

Multiply:

a. $ \displaystyle −1 \cdot 7$

b. $\displaystyle -1(-11)$

Solution (click to reveal)

Steps
 Algebraic
a. Multiply, noting that the signs are different so the product is negative. −1 · 7 = -7
−7 is the opposite of 7.
b. Multiply, noting that the signs are the same so the product is positive. −1(−11) = 11
11 is the opposite of −11.

Divide Integers[5]

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

Multiplication of integers Division of integers
5 \times 3 = 15 15 \div 3 = 5
-5 \times 3 = -15 -15 \div 3 = -5
5 \times -3 = -15 -15 \div -3 = 5
-5 \times -3 = 15 15 \div -3 = -5

Division follows the same rules as multiplication!

For the division of two signed numbers, when the:

signs are same, the quotient is positive.

signs are different, the quotient is negative.

And remember that we can always check the answer of a division problem by multiplying.

For multiplication and division of two signed numbers:

If the signs are the same, the result is positive.

If the signs are different, the result is negative.

Same signs Result
Two positives Positive
Two negatives Positive
If the signs are the same, the result is positive.
Different signs Result
Positive and negative Negative
Negative and positive Negative
If the signs are different, the result is negative.

Try it! – Dividing Numbers

Divide:

a. $\displaystyle −27 \div 3$

b. $ \displaystyle −100 \div (−4)$

Solution (click to reveal)

Steps
 Algebraic
a. Divide. With different signs, the quotient is negative. −27 \div 3 = -9
b. Divide. With signs that are the same, the quotient is positive. -100 \div (-4) = 25

Identify Multiples of a Number[6]

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.

Multiples of 2:

$\displaystyle 2, 4, 6, 8, 10, 12, \dotsc$

$\displaystyle \textcolor{red}{2 \cdot 1, 2 \cdot 2, 2 \cdot 3, 2 \cdot 4, 2 \cdot 5, 2 \cdot 6, \dotsc}$

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

Multiples of 3:

$\displaystyle 3, 6, 9, 12, 15, 18, \dotsc$

$\displaystyle \textcolor{red}{3 \cdot 1, 3 \cdot 2, 3 \cdot 3, 3 \cdot 4, 3 \cdot 5, 3 \cdot 6, \dotsc}$

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
Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24
Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36
Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48
Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60
Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72
Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84
Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96
Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108

Multiple

A number is a multiple of n if it is the product of a counting number and n.

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.

Use Common Divisibility Tests[7]

Another way to say that 375 is a multiple of 5 is to say that 375 is divisible by 5. In fact, 375 ÷ 5 is 75, so 375 is 5 ⋅ 75. We can see that 10,519 is not a multiple 3. When we divided 10,519 by 3, we did not get a counting number, so 10,519 is not divisible by 3.

Divisible

If a number m is a multiple of n, then we say that m is divisible by n.

Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. The table below summarizes divisibility tests for some of the counting numbers between one and ten.

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 divisible by both 2 and 3
10 if the last digit is 0

Try it! – Divisibility Test

Determine whether 1,290 is divisible by 2, 3, 5, and 10.

Solution (click to reveal)

The table applies the divisibility tests to 1,290. In the far right column, we check the results of the divisibility tests by seeing if the quotient is a whole number.

Divisible by…? Test Divisible? Check
2 Is the last digit 0, 2, 4, 6, or 8? Yes. yes 1290 ÷ 2 = 645
3 Is the sum of digits divisible by 3?
1 + 2 + 9 + 0 = 12 Yes.		 yes 1290 ÷ 3 = 430
5 Is the last digit 5 or 0? Yes. yes 1290 ÷ 5 = 258
10 Is the last digit 0? Yes. yes 1290 ÷ 10 = 129

Thus, 1,290 is divisible by 2, 3, 5, and 10.

Determine whether 5,625 is divisible by 2, 3, 5, and 10.

Solution (click to reveal)

Apply the divisibility tests to 5,625 and tests the results by finding the quotients.

Divisible by…? Test Divisible? Check
2 Is the last digit 0, 2, 4, 6, or 8? No. no 5625 ÷ 2 = 2812.5
3 Is the sum of digits divisible by 3?
5 + 6 + 2 + 5 = 18 Yes.		 yes 5625 ÷ 3 = 1875
5 Is the last digit 5 or 0? Yes. yes 5625 ÷ 5 = 1125
10 Is the last digit 0? No no 5625 ÷ 10 = 562.5

Thus, 5,625 is divisible by 3 and 5, but not 2 or 10.

Find all the Factors of the Given Number[8]

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


The image shows the equation 8 times 9 equals 72. The 8 and 9 are labeled as factors and the 72 is labeled product.

Factor and Product

If a ⋅ b = m, then a and b are factors of m, and m is the product of a and b.

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

Doing the Manipulative Mathematics activity “Model Multiplication and Factoring” will help you develop a better understanding of multiplication and factoring.

For example, suppose a choreographer is planning a dance for a ballet recital. There are 24 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 24. The following table summarizes the different ways that the choreographer can arrange the dancers.

Number of Groups Dancers per Group Total Dancers
1 24 1 ⋅24 = 24
2 12 2 ⋅ 12 = 24
3 8 3 ⋅ 8 = 24
4 6 4 ⋅ 6 = 24
6 4 6 ⋅ 4 = 24
8 3 8 ⋅ 3 = 24
12 2 12 ⋅ 2 = 24
24 1 24 ⋅ 1 = 24

What patterns do you see in the table? Did you notice that the number of groups times the number of dancers per group is always 24? This makes sense, since there are always 24 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 24, which are:

1, 2, 3, 4, 6, 8, 12, 24

We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with 1. 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.

How To:

  1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
    • If the quotient is a counting number, the divisor and quotient are a pair of factors.
    • If the quotient is not a counting number, the divisor is not a factor.
  2. List all the factor pairs.
  3. Write all the factors in order from smallest to largest.

Try it! – Finding Factors

Find all the factors of 72.

Solution (click to reveal)

Divide 72 by each of the counting numbers starting with 1. If the quotient is a whole number, the divisor and quotient are a pair of factors.

Dividend Divisor Quotient Factors
72 1 72 1, 72
72 2 36 2, 36
72 3 24 3, 24
72 4 18 4, 18
72 5 14.4 –
72 6 12 6, 12
72 7 ~10.29 –
72 8 9 8, 9

The next line would have a divisor of 9 and a quotient of 8. The quotient would be smaller than the divisor, so we stop. If we continued, we would end up only listing the same factors again in reverse order. Listing all the factors from smallest to greatest, we have 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

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Key Concepts

  • Mathematical operations
    Operation Notation Say: The result is…
    Addition a + b a plus b the sum of a and b
    Multiplication a · b, (a)(b), (a)b, a(b) a times b the product of a and b
    Subtraction a − b a minus b the difference of a and b
    Division a ÷ b, a/b a divided by b the quotient of a and b
  • Equality Symbol
    • a = b is read as a is equal to b
    • The symbol = is called the equal sign.

Inequality

  • a < b is read a is less than b
  • a is to the left of b on the number line

    ..
  • a > b is read a is greater than b
  • a is to the right of b on the number line

    ..
Algebraic Notation Say
a = b a is equal to b
a ≠ b a is not equal to b
a < b a is less than b
a > b a is greater than b
a ≤ b a is less than or equal to b
a ≥ b a is greater than or equal to b

 

Addition of Positive and Negative Integers
5 + 3 -5 + (-3) -5 – 3 5 – (-3)
8 -8 -2 2
both positive, sum positive both negative, sum negative different signs, more negatives, sum negative different signs, more positives, sum positive

Property of Absolute Value: |n| ≥ 0 for all numbers. Absolute values are always greater than or equal to zero!

Subtraction of Positive and Negative Integers
5 – 3 = 2 -5 – (-3) = -2 -5 – 3 = -8 5 – (-3) = 8
5 positives 5 negatives 5 negatives 5 positives
take away 3 positives take away 3 negatives want to subtract 3 positives want to subtract 3 negatives
2 positives 2 negatives need neutral pairs need neutral pairs
  • Subtraction Property: Subtracting a number is the same as adding its opposite.
  • Factors If a ⋅ b = m, then a and b are factors of m, and m is the product of a and b.
  • Find all the factors of a counting number.
    1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
      1. If the quotient is a counting number, the divisor and quotient are a pair of factors.
      2. If the quotient is not a counting number, the divisor is not a factor.
    2. List all the factor pairs.
    3. Write all the factors in order from smallest to largest.
  • Determine if a number is prime.
    1. Test each of the primes, in order, to see if it is a factor of the number.
    2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
    3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.
  • Divisibility tests
    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 divisible by both 2 and 3
    10 if the last digit is 0
  1. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction ;Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction ↵
  2. Section material derived from Openstax Prealgebra: The Language of Algebra-Use the Language of Algebra ↵
  3. Section material derived from Openstax Prealgebra: Introduction to Integers-Introduction to Integers ↵
  4. Section material derived from Openstax Prealgebra: Introduction to Integers-Multiply and Divide Integers ↵
  5. Section material derived from Openstax Prealgebra: Introduction to Integers-Multiply and Divide Integers ↵
  6. Section material derived from Openstax Elementary Algebra: Introduction to the Language of Algebra-Find Multiples and Factors ↵
  7. Section material derived from Openstax Prealgebra: Introduction to Whole Numbers-Introduction to Whole Numbers ↵
  8. Section material derived from Openstax Prealgebra: Introduction to Whole Numbers-Find Multiples and Factors ↵

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