 Properties of Multiplication – Commutative, Distributive, Associative, Closure, Identity

Check out different properties of multiplication of whole numbers to solve the problems easily. We have given six properties of multiplication in the below article. They are Closure Property, Zero Property, Commutative Property, Associativity Property, Identity Property, and Distributive Property.

The multiplication of whole numbers refers to the product of two or more whole numbers. Applying multiplication operation and the properties of multiplication are given clearly explained with the examples in this article for you. Have a look at the complete concept and improve your preparation level easily. Closure Property of Whole Numbers

According to Closure Property of Whole Numbers, if two whole numbers a and b are multiplied then their resultant a × b is also a whole number. Therefore, whole numbers are closed under multiplication.

a × b is a whole number, for every whole number a and b.

Verification:
In order to verify the Closure Property of Whole Numbers, let us take a few pairs of whole numbers and multiply them.

For example:
Let us take the whole numbers and multiply them to verify the Closure Property of Whole Numbers.
(i) 7 × 8 = 56
(ii) 0 × 15 = 0
(iii) 12 × 14 = 168
(iv) 21 × 1 = 21

We find that the product is always a whole number.

Commutativity of Whole Numbers / Order Property of Whole Numbers

The commutative property of multiplication of whole numbers states that altering the order of operands or the whole numbers does not affect the result of the multiplication.

a × b = b × a, for every whole number a and b.

Verification:
In order to verify the Commutativity of Whole Numbers property, let us take a few pairs of whole numbers and multiply the numbers in different orders as shown below.

Examples:
(i) 7 × 8 = 56 and 8 × 7 = 56
Both multiplications get the same output.
Therefore, 7 × 8 = 8 × 7
(ii) 30 × 10 = 300 and 10 × 30 = 300
Both multiplications get the same output.
Therefore, 30 × 10 = 10 × 30
(iii) 14 × 13 = 182 and 13 × 14 = 182
Both multiplications get the same output.
Therefore, 14 × 13 = 13 × 14
(iv) 16 × 17 = 272 and 17 × 16 = 272
Both multiplications get the same output.
Therefore, 16 × 17 = 17 × 16
(V) 1235 × 334 = 412490 and 334 × 1235 = 412490
Both multiplications get the same output.
Therefore, 1235 × 334 = 334 × 1235
(vi) 21534 × 1429 = 30772086 and 1429 × 21534 = 30772086
Both multiplications get the same output.
Therefore, 21534 × 1429 = 1429 × 21534

We find that in whatever order we multiply two whole numbers, the product remains the same.

Multiplication by Zero/Zero Property of Whole Numbers

On multiplying any whole numbers by zero the result is always zero. In general, if a and b are two whole numbers then,
a × 0 = 0 × a = 0
The product of any whole number and zero is always zero.

Verification:
In order to verify the Zero Property of Whole Numbers, we take some whole numbers and multiply them by zero as shown below

Examples:
(i) 30 × 0 = 0 × 30 = 0
(ii) 2 × 0 = 0 × 2 = 0
(iii) 127 × 0 = 0 × 127 = 0
(iv) 0 × 0 = 0 × 0 = 0
(v) 144 × 0 = 0 × 144 = 0
(vi) 54791 × 0 = 0 × 54791 = 0
(vii) 62888 × 0 = 0 × 62888 = 0

From the above examples, the product of any whole number and zero is zero.

Multiplicative Identity of Whole Numbers / Identity Property of Whole Numbers

On multiplying any whole number by 1 the result obtained is the whole number itself. In general, if a and b are two whole numbers then,
a × 1 = 1 × a = a
Therefore 1 is the Multiplicative Identity of Integers.

Verification:
In order to verify Multiplicative Identity of Whole Numbers, we find the product of different whole numbers with 1 as shown below

Examples:
(i) 16 × 1 = 16 = 1 × 16
(ii) 1 × 1 = 1 = 1 × 1
(iii) 27 × 1 = 27 = 1 × 27
(iv) 127 × 1 = 127 = 1 × 127
(v) 3518769 × 1 = 3518769
(vi) 257394 × 1 = 257394
We see that in each case a × 1 = a = 1 × a.

The number 1 is called the multiplication identity or the identity element for multiplication of whole numbers because it does not change the value of the numbers during the operation of multiplication.

Associativity Property of Multiplication of Whole Numbers

The result of the product of three or more whole numbers is irrespective of the grouping of these whole numbers. In general, if a, b and c are three whole numbers then, a × (b × c) = (a × b) × c.

Verification:
In order to verify the Associativity Property of Multiplication of Whole Numbers, we take three whole numbers say a, b, c, and find the values of the expression (a × b) × c and a × (b × c) as shown below :

Examples:
(i) (3 × 4) × 5 = 12 × 5 = 60 and 3 × (4 × 5) = 3 × 20 = 60
Both multiplications get the same output.
Therefore, (3 × 4) × 5 = 3 × (4 × 5)
(ii) (1 × 7) × 2 = 7 × 2 = 14 and 1 × (7 × 2) = 1 × 14 = 14
Both multiplications get the same output.
Therefore, (1 × 7) × 2 = 1 × (7 × 2)
(iii) (2 × 9) × 3 = 18 × 3 = 54 and 2 × (9 × 3) = 2 × 27 = 54
Both multiplications get the same output.
Therefore, (2 × 9) × 3 = 2 × (9 × 3).
(iv) (2 × 1) × 3 = 2 × 3 = 6 and 2 × (1 × 3) = 2 × 3 = 6
Both multiplications get the same output.
Therefore, (2 × 1) × 3 = 2 × (1 × 3).
(v) (221 × 142) × 421 = 221 × (142 × 421)
Both multiplications get the same output.
(vi) (2504 × 547) × 1379 = 2504 × (547 × 1379)
Both multiplications get the same output.
We find that in each case (a × b) × c = a × (b × c).
Thus, the multiplication of whole numbers is associative.

Distributive Property of Multiplication of Whole Numbers / Distributivity of Multiplication over Addition of Whole Numbers

According to the distributive property of multiplication of whole numbers, if a, b and c are three whole numbers then, a× (b + c) = (a × b) + (a × c) and (b + c) × a = b × a + c × a

Verification:
In order to verify the Distributive Property of Multiplication of Whole Numbers, we take any three whole numbers a, b, c and find the values of the expressions a × (b + c) and a × b + a × c as shown below

Examples:
(i) 3 × (2 + 5) = 3 × 7 = 21 and 3 × 2 + 3 × 5 = 6 + 15 =21
Therefore, 3 × (2 + 5) = 3 × 2 + 3 × 5
(ii) 1 × (5 + 9) = 1 × 14 = 15 and 1 × 5 + 1 × 9 = 5 + 9 = 14
Therefore, 1 × (5 + 9) = 1 × 5 + 1 × 9.
(iii) 2 × (7 + 15) = 2 × 22 = 44 and 2 × 7 + 2 × 15 = 14 + 30 = 44.
Therefore, 2 × (7 + 15) = 2 × 7 + 2 × 15.
(iv) 50 × (325 + 175) = 50 × 3250 + 50 × 175
(v) 1007 × (310 + 798) = 1007 × 310 + 1007 × 798

Questions and Answers on Properties of Multiplication

(i) Number × 0 = __________
(ii) 64 × __________ = 64000
(iii) Number × __________ = Number itself
(iv) 3 × (7 × 9) = (3 × 7) × __________
(v) 4 × _________ = 8 × 4
(vi) 7 × 6 × 11 = 11 × __________
(vii) 72 × 10 = __________
(viii) 6 × 48 × 100 = 6 × 100 × __________

Solutions:
(i) Number × 0 = __________
Any number multiplied with 0 gives 0 output.
Therefore, the answer is 0 × Number.
(ii) 64 × __________ = 64000
The given numbers are 64 × __________ = 64000
To get the answer multiply 64 with 1000.
64 × 1000.
(iii) Number × __________ = Number itself
Any number multiplied with 1 gives Number itself.
Therefore, the answer is Number × 1 = Number itself.
(iv) 3 × (7 × 9) = (3 × 7) × __________
The given numbers are 3 × (7 × 9) = (3 × 7) × __________
From the Associativity Property of Multiplication of Whole Numbers, 3 × (7 × 9) = (3 × 7) × 9 gives the same output.
(v) 4 × _________ = 8 × 4
The given numbers are 4 × _________ = 8 × 4
From the Commutativity of Whole Numbers, 4 × 8 = 8 × 4 gives the same output.
(vi) 7 × 6 × 11 = 11 × __________
The given numbers are 7 × 6 × 11 = 11 ×
From the Associativity Property of Multiplication of Whole Numbers, 7 × (6 × 11) = 11 × (7 × 6) gives the same output.
Therefore, the answer is 7 × 6.
(vii) 72 × 10 = __________
The given numbers are 72 × 10 = __________
From the Commutativity of Whole Numbers, 72 × 10 = 10 × 72 gives the same output.
Therefore, the answer is 10 × 72.
(viii) 6 × 48 × 100 = 6 × 100 × __________
The given numbers are 6 × 48 × 100 = 6 × 100 ×
From the Associativity Property of Multiplication of Whole Numbers, 6 × 48 × 100 = 6 × 100 × 48 gives the same output.