Multiply and Divide Integers

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter 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.

We remember that $a \cdot b$ means add $a, b$ times. Here, we are using the model shown in Figure 1 just to help us discover the pattern.

This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.

Now consider what it means 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 as shown in Figure 2.

This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives. Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.

In both cases, we started with $15$ neutral pairs. In the case on the left, we took away $5, 3$ times and the result was $- 15 .$ To multiply $(−5) (−3),$ we took away $- 5, 3$ times and the result was $15 .$ So we found that

\begin{matrix} 5 \cdot 3 = 15 & −5 (3) = −15 \\ 5 (−3) = −15 & (−5) (−3) = 15 \end{matrix}

Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.

Multiplication of Signed Numbers

The sign of the product of two numbers depends on their signs.

Same signs	Product
•Two positives •Two negatives	Positive Positive

Different signs	Product
•Positive • negative •Negative • positive	Negative Negative

Multiply each of the following:

ⓐ $−9 \cdot 3$
ⓑ $−2 (−5)$
ⓒ $4 (−8)$
ⓓ $7 \cdot 6$

Solution

Worked example: Multiplication of a negative and positive integer, showing the problem, explanation, and final result.
ⓐ
	$–9 \cdot 3$
Multiply, noting that the signs are different and so the product is negative.	$–27$

Example demonstrating the multiplication of two negative numbers, showing that a positive product results when signs are the same.
ⓑ
	$–2 (–5)$
Multiply, noting that the signs are the same and so the product is positive.	$10$

Example showing multiplication of integers with different signs, yielding a negative product.
ⓒ
	$4 (–8)$
Multiply, noting that the signs are different and so the product is negative.	$–32$

Illustrates the multiplication of 7 and 6, showing the positive product and the reason for its sign.
ⓓ
	$7 \cdot 6$
The signs are the same, so the product is positive.	$42$

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.

\begin{matrix} −1 \cdot 4 & −1 (−3) \\ −4 & 3 \\ −4 is the opposite of 4 & 3 is the opposite of −3 \end{matrix}

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

Multiply each of the following:

ⓐ $−1 \cdot 7$
ⓑ $−1 (−11)$

Solution

Explains properties of negative numbers, covering multiplication with different signs and opposites, with mathematical examples.
ⓐ
The signs are different, so the product will be negative.	$−1 \cdot 7$
Notice that −7 is the opposite of 7.	$−7$

Demonstrates rules for multiplying negative numbers, showing negative times negative equals positive, and the concept of opposites.
ⓑ
The signs are the same, so the product will be positive.	$−1 (−11)$
Notice that 11 is the opposite of −11.	$11$

Divide Integers

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

\begin{matrix} 5 \cdot 3 = 15 so 15 \div 3 = 5 & −5 (3) = −15 so −15 \div 3 = −5 \\ (−5) (−3) = 15 so 15 \div (−3) = −5 & 5 (−3) = −15 so −15 \div −3 = 5 \end{matrix}

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

Division of Signed Numbers

The sign of the quotient of two numbers depends on their signs.

Same signs	Quotient
•Two positives •Two negatives	Positive Positive

Different signs	Quotient
•Positive & negative •Negative & positive	Negative Negative

Remember, you can always check the answer to a division problem by multiplying.

Divide each of the following:

ⓐ $−27 \div 3$
ⓑ $−100 \div (−4)$

Solution

Example of integer division with a detailed explanation and the resulting quotient.
ⓐ
	$–27 \div 3$
Divide, noting that the signs are different and so the quotient is negative.	$–9$

Demonstration of dividing negative integers, illustrating the problem, an explanatory step, and the final positive result.
ⓑ
	$–100 \div (–4)$
Divide, noting that the signs are the same and so the quotient is positive.	$25$

Just as we saw with multiplication, when we divide a number by $1,$ the result is the same number. What happens when we divide a number by $−1 ?$ Let’s divide a positive number and then a negative number by $−1$ to see what we get.

\begin{matrix} 8 \div (−1) & −9 \div (−1) \\ −8 & 9 \\ −8 is the opposite of 8 & 9 is the opposite of −9 \end{matrix}

When we divide a number by, $−1$ we get its opposite.

Divide each of the following:

ⓐ $16 \div (−1)$
ⓑ $−20 \div (−1)$

Solution

This table illustrates the division of 16 by -1, explaining the rule that dividing a number by -1 results in its opposite.
ⓐ
	$16 \div (–1)$
The dividend, 16, is being divided by –1.	$–16$
Dividing a number by –1 gives its opposite.
Notice that the signs were different, so the result was negative.

This table illustrates the rule and provides an example of dividing a number by -1 to obtain its opposite.
ⓑ
	$–20 \div (–1)$
The dividend, –20, is being divided by –1.	$20$
Dividing a number by –1 gives its opposite.

Notice that the signs were the same, so the quotient was positive.

Simplify Expressions with Integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

$Simplify: 7 (−2) + 4 (−7) - 6 .$

Solution

We use the order of operations. Multiply first and then add and subtract from left to right.

Step-by-step simplification of a mathematical expression involving multiplication, addition, and subtraction of integers.
	$7 (−2) + 4 (−7) −6$
Multiply first.	$−14 + (−28) −6$
Add.	$−42 - 6$
Subtract.	$−48$

Simplify:

ⓐ ${(−2)}^{4}$
ⓑ ${−2}^{4}$

Solution

The exponent tells how many times to multiply the base.

ⓐ The exponent is $4$ and the base is $−2 .$ We raise $−2$ to the fourth power.

Illustrates the step-by-step process of calculating (-2)^4, showing its expanded form and sequential multiplication.
	${(−2)}^{4}$
Write in expanded form.	$(−2) (−2) (−2) (−2)$
Multiply.	$4 (−2) (−2)$
Multiply.	$- 8 (−2)$
Multiply.	$16$

ⓑ The exponent is $4$ and the base is $2 .$ We raise $2$ to the fourth power and then take the opposite.

Step-by-step calculation of -2^4, showing the expansion and sequential multiplication to determine the final value.
	$- 2^{4}$
Write in expanded form.	$- (2 \cdot 2 \cdot 2 \cdot 2)$
Multiply.	$- (4 \cdot 2 \cdot 2)$
Multiply.	$- (8 \cdot 2)$
Multiply.	$- 16$

$Simplify: 12 - 3 (9 - 12) .$

Solution

According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.

Step-by-step solution demonstrating the order of operations for the expression 12 - 3(9 - 12).
	$12 - 3 (9 - 12)$
Subtract the parentheses first.	$12 - 3 (−3)$
Multiply.	$12 - (−9)$
Subtract.	$21$

Simplify: $8 (−9) \div {(−2)}^{3} .$

Solution

We simplify the exponent first, then multiply and divide.

This table demonstrates the step-by-step simplification of the mathematical expression 8(-9) ÷ (-2)^3, culminating in the result of 9.
	$8 (−9) \div {(−2)}^{3}$
Simplify the exponent.	$8 (−9) \div (−8)$
Multiply.	$−72 \div (−8)$
Divide.	$9$

$Simplify: −30 \div 2 + (−3) (−7) .$

Solution

First we will multiply and divide from left to right. Then we will add.

This table demonstrates the step-by-step evaluation of a mathematical expression following the order of operations.
	$−30 \div 2 + (−3) (−7)$
Divide.	$−15 + (−3) (−7)$
Multiply.	$−15 + 21$
Add.	$6$

Evaluate Variable Expressions with Integers

Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

$Evaluate 2 x^{2} - 3 x + 8 when x = −4 .$

Solution



Simplify exponents.
Multiply.
Subtract.
Add.

Keep in mind that when we substitute $−4$ for $x,$ we use parentheses to show the multiplication. Without parentheses, it would look like $2 \cdot {−4}^{2} - 3 \cdot −4 + 8 .$

$Evaluate 3 x + 4 y - 6 when x = −1 and y = 2 .$

Solution


Substitute $x = −1$ and $y = 2$ .
Multiply.
Simplify.

Translate Word Phrases to Algebraic Expressions

Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient.

Translate to an algebraic expression and simplify if possible: the product of $−2$ and $14 .$

Solution

The word product tells us to multiply.

Steps to translate a verbal expression into a mathematical product and simplify it.
	the product of $−2$ and $14$
Translate.	$(−2) (14)$
Simplify.	$−28$

Translate to an algebraic expression and simplify if possible: the quotient of $−56$ and $−7 .$

Solution

The word quotient tells us to divide.

Step-by-step solution demonstrating the translation and simplification of a verbal division problem.
	the quotient of −56 and −7
Translate.	$−56 \div (−7)$
Simplify.	$8$

Key Concepts

Multiplication of Signed Numbers
- To determine the sign of the product of two signed numbers:
  
  Same Signs Product
  
  Two positives
  Two negatives Positive
  Positive
  
  Different Signs Product
  
  Positive • negative
  Negative • positive Negative
  Negative
Division of Signed Numbers
- To determine the sign of the quotient of two signed numbers:
  
  Same Signs Quotient
  
  Two positives
  Two negatives Positive
  Positive
  
  Different Signs Quotient
  
  Positive • negative
  Negative • Positive Negative
  Negative
Multiplication by $−1$
- Multiplying a number by $−1$ gives its opposite: $−1 a = - a$
Division by $−1$
- Dividing a number by $−1$ gives its opposite: $a \div (−1) = −a$

Same Signs	Product
Two positives Two negatives	Positive Positive

Different Signs	Product
Positive • negative Negative • positive	Negative Negative

Same Signs	Quotient
Two positives Two negatives	Positive Positive

Different Signs	Quotient
Positive • negative Negative • Positive	Negative Negative

Practice Makes Perfect

Multiply Integers

In the following exercises, multiply each pair of integers.

$−4 \cdot 8$

Solution

−32

$−3 \cdot 9$

$−5 (7)$

Solution

−35

$−8 (6)$

$−18 (−2)$

Solution

36

$−10 (−6)$

$9 (−7)$

Solution

−63

$13 (−5)$

$−1 \cdot 6$

Solution

−6

$−1 \cdot 3$

$−1 (−14)$

Solution

14

$−1 (−19)$

Divide Integers

In the following exercises, divide.

$−24 \div 6$

Solution

−4

$−28 \div 7$

$56 \div (−7)$

Solution

−8

$35 \div (−7)$

$−52 \div (−4)$

Solution

13

$−84 \div (−6)$

$−180 \div 15$

Solution

−12

$−192 \div 12$

$49 \div (−1)$

Solution

−49

$62 \div (−1)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$5 (−6) + 7 (−2) −3$

Solution

−47

$8 (−4) + 5 (−4) −6$

$−8 (−2) −3 (−9)$

Solution

43

$−7 (−4) −5 (−3)$

${(−5)}^{3}$

Solution

−125

${(−4)}^{3}$

${(−2)}^{6}$

Solution

64

${(−3)}^{5}$

$- 4^{2}$

Solution

−16

$- 6^{2}$

$−3 (−5) (6)$

Solution

90

$−4 (−6) (3)$

$−4 \cdot 2 \cdot 11$

Solution

−88

$−5 \cdot 3 \cdot 10$

$(8 - 11) (9 - 12)$

Solution

9

$(6 - 11) (8 - 13)$

$26 - 3 (2 - 7)$

Solution

41

$23 - 2 (4 - 6)$

$−10 (−4) \div (−8)$

Solution

−5

$−8 (−6) \div (−4)$

$65 \div (−5) + (−28) \div (−7)$

Solution

−9

$52 \div (−4) + (−32) \div (−8)$

$9 - 2 [3 - 8 (−2)]$

Solution

−29

$11 - 3 [7 - 4 (−2)]$

${(−3)}^{2} −24 \div (8 - 2)$

Solution

5

${(−4)}^{2} - 32 \div (12 - 4)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

−2 x + 17 when

ⓐ $x = 8$
ⓑ $x = −8$

Solution

ⓐ 1
ⓑ 33

−5 y + 14 when

ⓐ $y = 9$
ⓑ $y = −9$

10 - 3 m when

ⓐ $m = 5$
ⓑ $m = −5$

Solution

ⓐ −5
ⓑ 25

18 - 4 n when

ⓐ $n = 3$
ⓑ $n = −3$

$p^{2} - 5 p + 5 when p = −1$

Solution

11

$q^{2} - 2 q + 9$ when $q = −2$

$2 w^{2} - 3 w + 7$ when $w = −2$

Solution

21

$3 u^{2} - 4 u + 5$ when $u = −3$

$6 x - 5 y + 15$ when $x = 3$ and $y = −1$

Solution

38

$3 p - 2 q + 9$ when $p = 8$ and $q = −2$

$9 a - 2 b - 8$ when $a = −6$ and $b = −3$

Solution

−56

$7 m - 4 n - 2$ when $m = −4$ and $n = −9$

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

The product of $−3$ and 15

Solution

−3·15 = −45

The product of $−4$ and $16$

The quotient of $−60$ and $−20$

Solution

−60 ÷ (−20) = 3

The quotient of $−40$ and $−20$

The quotient of $−6$ and the sum of $a$ and $b$

Solution

$\frac{−6}{a + b}$

The quotient of $−7$ and the sum of $m$ and $n$

The product of $−10$ and the difference of $p and q$

Solution

−10 (p − q)

The product of $−13$ and the difference of $c and d$

Everyday Math

Stock market Javier owns $300$ shares of stock in one company. On Tuesday, the stock price dropped $$12$ per share. What was the total effect on Javier’s portfolio?

Solution

−$3,600

Weight loss In the first week of a diet program, eight women lost an average of $3 pounds$ each. What was the total weight change for the eight women?

Writing Exercises

In your own words, state the rules for multiplying two integers.

Solution

Sample answer: Multiplying two integers with the same sign results in a positive product. Multiplying two integers with different signs results in a negative product.

In your own words, state the rules for dividing two integers.

Why is ${−2}^{4} \neq {(−2)}^{4} ?$

Solution

Sample answer: In the first expression the base is positive and after you raise it to the power you should take the opposite. Then in the second expression the base is negative so you simply raise it to the power.

Why is ${−4}^{2} \neq {(−4)}^{2} ?$

Self Check

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

A self-assessment chart for math skills including multiplying and dividing integers, simplifying expressions, evaluating variable expressions, and translating word phrases to algebraic expressions.

ⓑ 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?