Multiplication and Division of Integers
Multiplication and division of integers are two basic operations that we perform on integers. Multiplication involves adding a particular number a given number of times. For example, 4 × 3 is nothing but adding 4 three times. On the other hand, division refers to distributing a quantity into equal groups. But, in the case of integers, we have to take care of the sign attached to numbers, which means whether a number is positive or negative. There are different rules for multiplication and division of integers that we are going to study in detail in this lesson.
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 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 can simply say integers. When we come to the case 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 x 5 = 10
When you multiply integers with two negative signs, Negative x Negative = Positive = –2 x –3 = 6
When you multiply integers with one negative sign and one positive sign, Negative x Positive = Negative = –2 x 5 = –10
The following table will help you remember rules for multiplying integers:
Types of Integers  Result  Example 
Both Integers Positive  Positive  2 x 5 = 10 
Both Integers Negative  Positive  –2 x –3 = 6 
1 Positive and 1 Negative  Negative  –2 x 5 = –10 
Anna eats 4 cookies per day. How many cookies does she eat in 5 days? ⇒ 5 × 4 = 20 cookies.
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 or dividing 2 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 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 multiplication and division of integers in detail.
Closure Property
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 
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 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
According to the associative property, changing the grouping of two integers does not alter the result of the operation. The associative property applies to the addition and multiplication of two integers but not in case of 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
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.
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
In case of addition of integers, 1 is the multiplicative identity. There is no identity element in case of 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 
Important Notes
 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.
Solved Examples

Example 1: Divide the given expression: (–20) ÷ (–5) ÷ (–2) = ?
Solution:
Here, we have to divide three integers, so we will follow BODMAS rule here as there is more than one operation in this expression. Step 1 is (–20 ÷ –5) ÷ (–2). Now, by dividing 20 by 5, we get 4 as the answer. So, the new expression is (4) ÷ (–2). 4 is a positive integer, as negative ÷ negative = positive. 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:
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.
Practice Questions
FAQs on Multiplication and Division of Integers
What are Integers?
The positive integers, zero, and the negative numbers together form integers.
How to Add and Subtract Integers?
If two integers are to be added or subtracted, look at the signs of the integers. If both the numbers are positive, add them and the sum is positive. If both the numbers are negative, add them and the sum is negative. If either of the numbers is positive or negative, subtract and the difference will have the sign of the bigger number.
Can Integers be Negative?
Yes, integers include all the negative numbers. They are the number opposites of the positive numbers. Example: 5, 63, 5, 1, and so on.
Are Integers Whole Numbers?
Yes. All the whole numbers 0, 1, 2, 3,........ are integers. The set of whole numbers belongs to the set of integers.
What is the Rule of Division of Integers?
The rules for division of integers are given below:
 Positive ÷ positive = positive
 Negative ÷ negative = positive
 Negative ÷ positive = negative
How do you Multiply Integers?
While multiplying, follow this trick to easily get the answer:
 Multiply without the negative sign.
 If both the integers are negative or both are positive, the final answer will be positive.
 If one integer is positive and the other is negative, the final answer 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 x Positive = Positive
 Rule 2: Positive x Negative = Negative
 Rule 3: Negative x Positive = Negative
 Rule 4: Negative x 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.