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
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