Multiplication and Division of Integers
The multiplication and division of integers are two of the basic operations performed on integers. Multiplication of integers is the same as the repetitive addition which means adding an integer a specific number of times. For example, 4 × 3 means adding 4 three times, i.e 4 + 4 + 4 = 12. Division of integers means equal grouping or dividing an integer into a specific number of groups. For example, 6 ÷ 2 means dividing 6 into 2 equal parts, which results in 3. Let us learn more about the multiplication and division of integers in this article.
What is Multiplication and Division of Integers?
The four basic arithmetic operations associated with integers are:
 Addition of integers
 Subtraction of integers
 Multiplication of integers
 Division of integers
Multiplication and division of integers are the most important arithmetic operations used often. Let us learn the multiplication and division of integers in detail.
Multiplication of Integers
Multiplication of integers is the process of repetitive addition including positive and negative numbers or we can simply say integers. When we come to the case multiplication of integers, the following cases must be taken into account:
 Multiplying 2 positive numbers
 Multiplying 2 negative numbers
 Multiplying 1 positive and 1 negative number
When you multiply integers with two positive signs, Positive x Positive = Positive = 2 × 5 = 10.
When you multiply integers with two negative signs, Negative x Negative = Positive = –2 × –3 = 6.
When you multiply integers with one negative sign and one positive sign, Negative x Positive = Negative = –2 × 5 = –10.
The following table will help you remember rules for multiplying integers:
Types of Integers  Result  Example 

Both Integers Positive  Positive  2 × 5 = 10 
Both Integers Negative  Positive  –2 × –3 = 6 
1 Positive and 1 Negative  Negative  –2 × 5 = –10 
Example: Anna eats 4 cookies per day. How many cookies does she eat in 5 days? ⇒ 5 × 4 = 20 cookies.
Multiplication of Integers Rules and Steps
Multiplication of integers is very similar to normal multiplication. However, since integers deal with both negative and positive numbers, we have certain rules or conditions to remember while multiplying integers as we saw in the previous section. Let us look at the steps for multiplying integers.
 Step 1: Determine the absolute value of the numbers.
 Step 2: Find the product of the absolute values.
 Step 3: Once the product is obtained, determine the sign of the number according to the rules or conditions.
Let us look at an example to understand the steps better. Multiply  7 × 8.
Step 1: Determine the absolute value of  7 and 8.
7 = 7 and 8 = 8.
Step 2: Find the product of the absolute value numbers 7 and 8.
7 × 8 = 56
Step 3: Determine the sign of the product according to the multiplication of integers rules. According to the multiplication of integer rule, if a negative number is multiplied with a positive number, then the product is a negative number.
Therefore,  7 × 8 =  56.
Division of Integers
Division of integers involves the grouping of items. It includes both positive numbers and negative numbers. Just like multiplication, the division of integers also involves the same cases.
 Dividing 2 positive numbers
 Dividing 2 negative numbers
 Dividing 1 positive and 1 negative number
When you divide integers with two positive signs, Positive ÷ Positive = Positive → 16 ÷ 8 = 2.
When you divide integers with two negative signs, Negative ÷ Negative = Positive → –16 ÷ –8 = 2.
When you divide integers with one negative sign and one positive sign, Negative ÷ Positive = Negative → –16 ÷ 8 = –2.
The following table will help you remember rules for dividing integers:
Types of Integers  Result  Example 

Both Integers Positive  Positive  16 ÷ 8 = 2 
Both Integers Negative  Positive  –16 ÷ –8 = 2 
1 Positive and 1 Negative  Negative  –16 ÷ 8 = –2 
To sum it all up and to make everything easy, the two most important things to remember when you are multiplying integers or dividing integers are:
 When the signs are different, the answer is always negative.
 When the signs are the same, the answer is always positive.
Multiplication and Division of Integers Examples
Few examples of multiplication and division of integers are shown in the table given below:
Multiplication  Division 

4 × 2 = 8  15 ÷ 3 = 5 
4 × 2 = 8  15 ÷ –3 = –5 
4 × 2 = 8  –15 ÷ 3 = –5 
4 × 2 = 8  –15 ÷ –3 = 5 
Properties of Multiplication and Division of Integers
Multiplication and division of integers properties help us to identify the relationship between two or more integers when they are linked by multiplication or division operation between them. There are a few properties associated with the multiplication and division of integers.
Properties related to multiplication and division of integers are listed below:
 Closure Property
 Commutative Property
 Associative Property
 Distributive Property
 Identity Property
Let's understand each property in relation to the division and multiplication of integers in detail.
Closure Property of Multiplication of Integers
The closure property states that the set is closed for any particular mathematical operation. Integers are closed under addition, subtraction, and multiplication. However, they are not closed under division.
Operation  Example 

a × b is an integer  2 × –6= –12 
a ÷ b not always an integer  –3/4 is a fraction 
Multiplication of Integers Commutative Property
According to the commutative property, interchanging the positions of operands in an operation does not affect the result. The addition and multiplication of integers follow the commutative property, while the division of integers does not hold this property.
Operation  Example 

a × b = b × a  5 × (–6) and (–6) × 5 = –30 
a ÷ b ≠ b ÷ a  15 ÷ 3 = 5 but 3 ÷ 15 = 1/5 
Associative Property of Multiplication of Integers
According to the associative property, changing the grouping of integers does not alter the result of the operation. The associative property applies to the addition and multiplication of two integers but not in the case of the division of integers.
Operation  Example 

(a × b) × c = a × (b × c)  (5 × –3) × 2 = –30 5 × (–3 × 2) = –30 
(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)  (20 ÷ 5) ÷ 2 = 2 but 20 ÷ (5 ÷ 2)= 8 
Distributive Property of Multiplication of Integers
Distributive property states that for any expression of the form a (b + c), which means a × (b + c), operand a can be distributed among operands b and c as (a × b + a × c) i.e., a × (b + c) = a × b + a × c. Multiplication of integers is distributive over addition and subtraction. The distributive property does not hold true for the division of integers.
Operation  Example 

a × (b + c) = (a × b) + (a × c)  4 × (–3 + 6) =12 (4 × –3) + (4 × 6) = 12 
a × (b – c) = (a × b) – (a × c)  2 × (5 – 3) = 4 (2 × 5) – (2 × 3) = 4 
Identity Property of Multiplication of Integers
In the case of the multiplication of integers, 1 is the multiplicative identity. There is no identity element in the case of the division of integers.
Identity Under Addition is 0  Identity Under Multiplication is 1 

For any integer a, a + 0 = 0 + a = a  For any integer a, 1 × a = a × 1 = a 
For example, 8 + 0 = 0 + 8 = 8  For example, (– 4) × 1 = 1 × (– 4) = – 4 
Multiplication and Division of Integers Tips and Tricks:
 There is neither the smallest integer nor the biggest integer.
 The smallest positive integer is 1 and the greatest negative integer is 1.
 PEMDAS rule applies for operations on integers. “Operations” are any of the following: Brackets, Squares, Powers, Square Roots, Division, Multiplication, Addition, and Subtraction.
Related Articles:
Check these interesting articles related to the concept of multiplication and division of integers.
Examples of Division and Multiplication of Integers

Example 1: Solve the given expression by using the division of integers rules: (–20) ÷ (–5) ÷ (–2).
Solution:
Here, we have to divide three integers, so we will follow the BODMAS rule here as there is more than one operation in this expression. The first step is (–20 ÷ –5). Now, by dividing 20 by 5, we get 4 as the answer. 4 is a positive integer, as negative ÷ negative = positive. So, the new expression is (4) ÷ (–2). Now, if we divide 4 by 2, we get 2, as positive ÷ negative = negative. Therefore, (–20) ÷ (–5) ÷ (–2) = –2.

Example 2: A test has 20 questions. Correct answers get +3 and incorrect answers 1. A student answered 5 questions incorrectly. How many points did the student score?
Solution:
To solve this question, we will be using the concept of rules for the multiplication of integers. If 1 answer is correct, 3 points are awarded. Thus, for 15 correct answers, points acquired will be 15 × 3 = 45. If 1 answer is incorrect, 1 point is given. Thus, for 5 incorrect answers, points will be (5 × 1) = 5. Subsequently, the points scored by the student will be 45  5 = 40. Therefore, the final score is 40 points.

Example 3: Help Jane solve the equation by using one of the properties of multiplication of integers. (35) × (101).
Solution:
Given, (35) × (101), which can rewritten as (35) × (100 + 1). By using the distributive property,
(35 × 100) + (35 × 1)
= 3500 + 35
= 3535
Therefore, the value of (35) × (101) = 3535.
FAQs on Multiplication and Division of Integers
What is Multiplication of Integers?
Multiplication of integers is the repetitive addition of numbers which means that a number is added to itself a specific number of times. For example, 4 × 2, which means 4 is added two times. This implies, 4 + 4 = 4 × 2 = 8.
What are the Properties of Multiplication of Integers with Examples?
The properties of multiplication of integers are given below:
 Closure property → 2 × 3 = 6, where 2, 3, and 6 are integers.
 Associative property → (2 × 3) × (9) = 2 × (3 × 9) = 54.
 Commutative property → 4 × 7 = 7 × 4 = 28.
 Distributive property → 3 × (4 + 2) = (3 × 4) + (3 × 2) = 6.
 Identity element → 3 × 1 = 1 × 3 = 3. 1 is the identity element.
What are the Rules for Multiplication and Division of Integers?
The basic rules for division and multiplication of integers are given below:
 Multiplication or division of two numbers with the same sign results in a positive number.
 Multiplication or division of two numbers with opposite signs results in a negative number.
What are the Properties of Division of Integers?
The properties of the division of integers are given below:
 If we divide 0 by any nonzero integer, the answer will always be 0. It can be mathematically expressed as 0 ÷ a = 0.
 Any integer divided by itself results in 1. This implies, a ÷ a = 1.
 When an integer is divided by another integer, then it satisfies the division algorithm which says 'dividend = divisor × quotient + remainder'.
 When an integer is divided by 1, the result is always the integer itself. For example, 5 ÷ 1 = 5.
What is the Rule of Division of Integers?
The rules for the division of integers are given below:
 Positive ÷ positive = positive
 Negative ÷ negative = positive
 Negative ÷ positive = negative
How do you Multiply Integers?
While multiplying integers, follow this trick to easily get the answer:
 Multiply without the negative sign.
 If both the integers are negative or both are positive, the product will be positive.
 If one integer is positive and the other is negative, the product will be negative.
How do you Multiply Multiple Integers?
If there are more than two integers, then follow these simple steps to multiply them:
 Multiply without the negative sign.
 The sign of the final answer can be determined by the number of negative signs.
 If the total number of negative signs is even, the final answer will be positive.
 If the total number of negative signs is odd, the final answer will be negative.
What are the Four Rules for Multiplying Integers?
Four rules of multiplying integers are stated below:
 Rule 1: Positive × Positive = Positive
 Rule 2: Positive × Negative = Negative
 Rule 3: Negative × Positive = Negative
 Rule 4: Negative × Negative = Positive
How do you Multiply a Positive and Negative Integer?
When we have two integers, one positive and one negative, follow these simple steps to get their product:
 Multiply without the negative sign.
 Add the negative sign to the answer to get the final answer.
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