Multiplication of Integers Properties, Rules, and Formulae are here. Follow various tips and shortcuts of Integers Multiplication. Explore the various properties of Integers when multiplying. Know the importance and application of each property rule. Go through the below sections to know more about how multiplication affects integer values. Also, find the example questions and procedures to find solutions.

## Properties of Multiplication of Integers

The multiplication of integers refers to the product of 2 or 3 integer values. The set of numbers which consists of whole numbers, natural numbers, the additive inverse, and zero are known as integers. Therefore, integers consist of positive or negative and they have sign or magnitude associated with them. Integers are represented with the letter “Z”. In the below sections, you can get the complete information regarding the properties of the multiplication of integers. Moreover, you can also find the solved examples, rules, definitions, formulae, etc.

Before knowing the properties, get perfection in all the fields of Integers. In the previous articles, you can find the important concepts of Integer values and also various operations. Various arithmetic operations can be performed on Integers like addition, subtraction, multiplication, division, etc. Each operation possesses several properties and here we are going to discuss various properties of multiplication.

### Integers Multiplication Properties

There are 6 properties of integers multiplication. Each property will be discussed in detail in the further sections. The important properties are

- Closure Property of Multiplication
- Commutative Property of Multiplication
- Associative Property of Multiplication
- Distributive Property of Multiplication
- Multiplication by zero
- Multiplicative Identity

To carry out multiplication, you need to always multiply the exact integer values and there are a few rules to keep in mind to determine the sign of the final result.

#### 1. Closure Property of Multiplication

According to the closure property, if two positive or negative integers are multiplied, then the result is also an integer.

Suppose two integers are x and y, then the product xy is also an integer.

**Example:**

7*5 = 35

5*(-2)=-10

(-2)*(-5)= 10

In the above example, we can change the operands of 7 and 5, in either way we get the same result. We can multiply 7 with 5 or we can multiply 5 with 7, we get the same result.

#### 2. Commutative Property of Multiplication

According to the commutative property, altering the integers or operands order does not affect the result of the multiplication.

x*y=y*x, for every integer x and y

**Example:**

7*(-3)=-(7*3)= -21

(-5)*(-6)=(-6)*(-5)=30

In the above example, we can change the operands or sign. We can multiply 7 with -3 or we can multiply 7 and 3 and add a negative sign to it.

#### 3. Associative Property of Multiplication

According to the associative property, the grouping of integers can be changed and it does not affect the final result.

Suppose x, y and z are integers

x*(y*z)=(x*y)*z

**Example:**

4*(2*5)=(4*2)*5=40

2*(5*3)=(2*5)*3=30

As shown in the above example, we can multiply 2 and 5, then multiply the product with 4 to get the final result or we can multiply 4 and 2, then multiply the product with 5 to get the final answer.

#### 4. Distributive Property of Multiplication

According to the distributive property of multiplication, it explains the ability to distribute an operation over other mathematical operations within the bracket. This property can be of distributive property of multiplication over addition or distributive property of multiplication over subtraction.

Suppose x,y and z are integers

x*(y+z)=(x*y)=(x*z)

**Example:**

2*(3+5)=(2*3)+(2*5)=16

4*(4+4)=(4*4)+(4*4)=32

In the above example, we can first add 3 and 5 and then divide it by 2 or we can multiply 2 and 3, then multiply 2 and 5, then add both the products to get the final answer.

#### 5. Multiplication by zero

According to multiplication by zero, any integer multiplied by 0 results in zero

Suppose x is an integer

x*0=0

**Example:**

2*0=0

5*0=0

In the above example, the integer value 2 is multiplied by 0 and the result is 0.

#### 6. Multiplicative Identity

According to the multiplicative identity, any integer multiplied by 1 results in the same integer value.

Suppose x is an integer

x*1=x

**Example:**

4*1=4

6*1=6

In the above example, the integer 4 is multiplied by 1 and the result is 4.

### Key Points to Remember about Integers Multiplication Properties

- Closure Property of integer multiplication refers to that the product of two or three integer values will be an integer.
- The commutative property of multiplication refers to that altering two or three integer numbers will not change the value of the final answer
- The associative property of multiplication refers that the way the grouping of integer values together does not affect the final answer.
- The distributive property of multiplication refers to the distribution procedure of one operation value over other mathematical integer values within the braces.
- Multiplication by zero refers that the product of any negative or positive integer by zero
- Multiplicative Integer refers to the final answer as 1 when any integer is multiplied with 1.

### Solved Examples on Properties of Multiplication of Integers

**Question 1.**

Adam is Scuba diving. He descends 5 feet. He descends the same distance 4 more times. How many feet is Adam’s new distance from the sea level?

**Solution:**

As given in the question,

Adam descends 5 feet, descends means traveling down. Therefore, it is negative

Also given, he descends 4 more times. Initially, he descended once and again 4 times. Therefore, it is 5 times.

Hence, (-5)*5=-25

The solution is Adam’s new distance is 25 feet from the sea level.

**Question 2.**

Clark made four of his trucks payments late and was fined four late fees. The total change to his savings account was -$28. What integer represents one late fee?

**Solution:**

As given in the question, Clark made four truck payments late which means he did 4 payments. It is positive

Also given, his savings account balance is -$28.

Amount of each payment = -28/4 = -$7

Therefore, each payment for one late fee is $7.

The solution is $7.

**Question 3.**

Jan lost 22 points on her exam. She got 11 questions wrong. What integer represents the point value of each wrong answer?

**Solution:**

As given in the question, Jan lost 22 points. Therefore it is negative.

To find the points lost for each wrong answer = 22/11 = 2

Therefore, she lost 2 points for each wrong answer.

**Question 4.**

During the field trip to the zoo, the class was divided into 8 groups of 3. How many students are there in the class?

**Solution:**

As given in the question,

The class was divided into 8 groups of 3

Therefore, the total number of students in the class are 8*3=24

The solution is 24.

**Question 5.**

You are making a snack for 4 friends. Each need 4 crackers. How many crackers do you need?

**Solution:**

As given in the question,

There are 4 friends and each needs 4 crackers to prepare a snack.

Therefore, the total number of crackers you need are 4*4=16

The solution is 16 crackers.

In the above article, we have provided all the information regarding Properties of Multiplication of Integers, its rules, solved examples, procedure, etc. Before solving the questions, know about the various properties and get perfect in them. Hope you liked the article and this will help you in your preparation. If you need any further clarifications, contact us through the comment section or directly message us. Bookmark our page for all the latest updates.