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: *

*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.

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).

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)

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

For every integer a, we have

a × 1 = a = 1 × a

The integer 1 is called the multiplicative identity for integers.

For any integer, we have

a × 0 = 0 = 0 × a

**For example: **

(i) m × 0 = 0

(ii) 0 × y = 0

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.

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

**Properties of Multiplication of Integers**

**Examples on Multiplication of Integers**

**Properties of Division of Integers**

**Examples on Division of Integers**

**Examples on Fundamental Operations**

● Numbers - Worksheets

**Worksheet on Multiplication of Integers**

**Worksheet on Division of Integers**

**Worksheet on Fundamental Operation**

**7th Grade Math Problems**** ****From Properties of Multiplication of Integers to HOME PAGE**

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