# Properties of Multiplication of Integers

The properties of multiplication of integers are discussed with examples. All properties of multiplication of whole numbers also hold for integers.

The multiplication of integers possesses the following properties:

### Property 1 (Closure property):

The product of two integers is always an integer.

That is, for any two integers m and n, m x n is an integer.

For example:

(i) 4 × 3 = 12, which is an integer.

(ii) 8 × (-5) = -40, which is an integer.

(iii) (-7) × (-5) = 35, which is an integer.

### Property 2 (Commutativity property):

For any two integer’s m and n, we have

m × n = n × m

That is, multiplication of integers is commutative.

For example:

(i) 7 × (-3) = -(7 × 3) = -21 and (-3) × 7 = -(3 × 7) = -21

Therefore, 7 × (-3) = (-3) × 7

(ii) (-5) × (-8) = 5 × 8 = 40 and (-8) × (-5) = 8 × 5 = 40

Therefore, (-5) × (-8) = (-8) × (-5).

### Property 3 (Associativity property):

The multiplication of integers is associative, i.e., for any three integers a, b, c, we have

a × ( b × c) = (a × b) × c

For example:

(i) (-3) × {4 × (-5)} = (-3) × (-20) = 3 × 20 = 60

and, {(-3) × 4} × (-5) = (-12) × (-5) = 12 × 5 = 60

Therefore, (- 3) × {4 × (-5)} = {(-3) × 4} × (-5)

(ii) (-2) × {(-3) × (-5)} = (-2) × 15 = -(2 × 15)= -30

and, {(-2) × (-3)} × (-5) = 6 × (-5) = -(6 × 5) = -30

Therefore, (- 2) × {(-3) × (-5)} = {-2) × (-3)} × (-5)

### Property 4 (Distributivity of multiplication over addition property):

The multiplication of integers is distributive over their addition. That is, for any three integers a, b, c, we have

(i) a × (b + c) =a × b + a × c

(ii) (b + c) × a = b × a + c × a

For example:

(i) (-3) × {(-5) + 2} = (-3) × (-3) = 3 × 3 = 9

and, (-3) × (-5) + (-3) × 2 = (3 × 5 ) -( 3 × 2 ) = 15 - 6 = 9

Therefore, (-3) × {(-5) + 2 } = ( -3) × (-5) + (-3) × 2.

(ii) (-4) × {(-2) + (-3)) = (-4) × (-5) = 4 × 5 = 20

and, (-4) × (-2) + (-4) × (-3) = (4 × 2) + (4 × 3) = 8 + 12 = 20

Therefore, (-4) × {-2) + (-3)} = (-4) × (-2) + (-4) × (-3).

Note: A direct consequence of the distributivity of multiplication over addition is

a × (b - c) =a × b - a × c

### Property 5 (Existence of multiplicative identity property):

For every integer a, we have

a × 1 = a = 1 × a

The integer 1 is called the multiplicative identity for integers.

### Property 6 (Existence of multiplicative identity property):

For any integer, we have

a × 0 = 0 = 0 × a

For example:

(i) m × 0 = 0

(ii) 0 × y = 0

### Property 7:

For any integer a, we have

a × (-1) = -a = (-1) × a

Note: (i) We know that -a is additive inverse or opposite of a. Thus, to find the opposite of inverse or negative of an integer, we multiply the integer by -1.

(ii) Since multiplication of integers is associative. Therefore, for any three integers a, b, c, we have

(a × b) × c = a × (b × c)

In what follows, we will write a × b × c for the equal products (a × b) × c and a × (b × c).

(iii) Since multiplication of integers is both commutative and associative. Therefore, in a product of three or more integers even if we rearrange the integers the product will not change.

(iv) When the number of negative integers in a product is odd, the product is negative.

(v) When the number of negative integers in a product is even, the product is positive.

### Property 8

If x, y, z are integers, such that x > y, then

(i) x × z > y × z, if z is positive

(ii) x × z < y × z , if z is negative.

These are the properties of multiplication of integers needed to follow while solving the multiplication of integers.

Numbers - Integers

Integers

Multiplication of Integers

Properties of Multiplication of Integers

Examples on Multiplication of Integers

Division of Integers

Absolute Value of an Integer

Comparison of Integers

Properties of Division of Integers

Examples on Division of Integers

Fundamental Operation

Examples on Fundamental Operations

Uses of Brackets

Removal of Brackets

Examples on Simplification

Numbers - Worksheets

Worksheet on Multiplication of Integers

Worksheet on Division of Integers

Worksheet on Fundamental Operation

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