# Properties of Multiplication of Integers

Properties of  Multiplication of Integers: Closure Property, Commutative Property, Associative Property, Identity Property and Distributive Property are explained with examples.

Properties of multiplication

(1)  Closure property-

Multiplication of any two integers gives integers. It is known as integers are closed under multiplication.

Condition for closure property-

x × y = z

Here, x, y and z are integers.

Eg:-

(i) 5× 3 = 15

(ii) 6 × 3 = 18

Hence, multiplication is closure for integers.

(2) Commutative property-

Integers can be multiplied in any order.

Condition for commutative property-

x × y = y × x

Here, x and y are integers.

Eg:-

(i)

5 × 7 =7× 5

35 = 35

(ii)

6 × 9 = 9 × 6

54 = 54

In both examples, we are getting same result after multiplication in both sides.

Hence, multiplication is commutative for integers.

(3) Multiplication by zero-

Product of an integer and zero is zero.

(i) 3 × 0 = 0

(ii) 7 × 0 = 0

(4) Associative property

Integers can be grouped differently in multiplication.

Condition for associative property:

x × (y ×  z) = (y × x) × z

Here, x, y and z are integers.

Eg:-

(i) 5 × (6 × 8 ) can be grouped as (5 × 6) × 8

Now, check the total

5 × (6 × 8) = 240 and

(5 × 6) × 8= 240

Results are same.

(ii) -3 × (2 × 5) = (-3 × 2) × 5

-30 = -30

Results are same.

Hence, multiplication is associative for integers.

(5)Identity property-

Condition for multiplicative identity-

Multiplicative identity of a whole number states that

x × identity = x = identity × x

Here, x is a whole number.

Eg:-

(i) -7 × 1 = – 7

(ii) 6 × 1= 6

In both examples, when we multiply 1 we get same integer.

Hence, one is a multiplicative identity for integer.

(6)Distributive property

Whole numbers under multiplication have distributive property. Distributive property exists only under multiplication operation.

Because, distributive property can only exists when multiplication and addition both involve in an expression. This property is also known as ‘distributive property of multiplication over addition`. It tells us that we distribute the multiplication over all the terms inside the parenthesis or brackets by multiply terms with terms of brackets.

Condition for distributive property of multiplication over addition:

a ×  (b + c) = ab + ac

a is distribute to b and c by multiplying a inside the terms of brackets.

Eg:-

(i) 2 × (7+8) =2 × 7+ 2 × 8

2 ×15 =14 + 16

30 = 30

(ii) 3 × (5 + 8) =3 × 5 + 3 × 8

39 =15 + 24

39 = 39

In both examples, results are same.

Hence, distributive property can be executed in multiplication.

Making Multiplication Easier-

In this method we use commutative and associative property of integers.

Eg1- Find 16×14

Method

As in above example

14=10+4

We can write

16× (10+4)

=16×10+16×4 [Associativity of integers]

=160+64

=224

Eg2 – Find (-25)×37×4

Method

We can arrange them as

-25×4×37[Commutativity of integers]

=100×37

=3700

Helping Topics

Integers