There are six properties of multiplication of entire numbers that

will assist to unravel the issues simply.

The six properties of multiplication are Closure Property, Commutative Property,

Zero Property, Id Property,

Associativity Property and Distributive

Property.

The properties of multiplication on entire numbers are mentioned under; these properties will assist us find the product of even very giant numbers conveniently.

Closure Property of Complete Numbers:

If a and b are two numbers, then their product a × b can be a complete quantity.

In different phrases, if we multiply two entire numbers, we get a complete quantity.

**Verification:**

With a view to confirm this property, allow us to take a couple of pairs of entire numbers and multiply them;

**For instance:**

(i) 8 × 9 = 72

(ii) 0 × 16 = 0

(iii) 11 × 15 = 165

(iv) 20 × 1 = 20

We discover that the product is all the time a complete numbers.

Commutativity of Complete Numbers / Order Property of Complete Numbers:

The multiplication of entire numbers is commutative.

In different phrases, if a and b are any two entire numbers, then a × b = b × a.

We are able to multiply numbers in any order. The product doesn’t

change when the order of numbers is modified.

When multiplying

any two numbers, the product stays similar whatever the order of

multiplicands. We are able to multiply numbers in any order, the product stays the

similar.

For Instance:

(i) 7 × 4 = 28

(ii) 4 × 7 = 28

**Verification:**

With a view to confirm this property, allow us to take a couple of pairs of entire numbers and multiply these numbers in numerous orders as proven under;

**For Instance:**

(i) 7 × 6 = 42 and 6 × 7 = 42

Subsequently, 7 × 6 = 6 × 7

(ii) 20 × 10 = 200 and 10 × 20 = 200

Subsequently, 20 × 10 = 10 × 20

(iii) 15 × 12 = 180 and 12 × 15 = 180

Subsequently, 15 × 12 = 12 × 15

(iv) 12 × 13 = 156 and 13 × 12

Subsequently, 12 × 13 = 13 × 12

(V) 1122 × 324 = 324 × 1122

(vi) 21892 × 1582 = 1582 × 21892

We discover that in no matter order we multiply two entire numbers, the product stays the identical.

III. Multiplication By Zero/Zero Property of Multiplication of Complete Numbers:

When a quantity is multiplied by 0, the product is all the time 0.

If a is any entire quantity, then a × 0 = 0 × a = 0.

In different phrases, the product of any entire quantity and nil is all the time zero.

When 0 is multiplied by any quantity the

product is all the time zero.

For instance:

(i) 3 × 0 = 0 + 0 + 0 = 0

(ii) 9 × 0 = 0 + 0 + 0 = 0

**Verification:**

With a view to confirm this property, we take some entire numbers and multiply them by zero as proven under;

**For instance:**

(i) 20 × 0 = 0 × 20 = 0

(ii) 1 × 0 = 0 × 1 = 0

(iii) 115 × 0 = 0 × 115 = 0

(iv) 0 × 0 = 0 × 0 = 0

(v) 136 × 0 = 0 × 136 = 0

(vi) 78160 × 0 = 0 × 78160 = 0

(vii) 51999 × 0 = 0 × 51999 = 0

We observe that the product of any entire quantity and nil is zero.

IV. Multiplicative Id of Complete Numbers / Id Property of Complete Numbers:

When a quantity is multiplied by 1, the product is the quantity

itself.

If a is any entire quantity, then a × 1 = a = 1 × a.

In different phrases, the product of any entire quantity and 1 is the quantity itself.

When 1 is multiplied by any quantity the

product is all the time the quantity itself.

For instance:

(i) 1 × 2 = 1 + 1 = 2

(ii) 1 × 6 = 1 + 1 + 1 + 1 + 1 + 1 = 6

**Verification:**

With a view to confirm this property, we discover the product of various entire numbers with 1 as proven under:

**For instance:**

(i) 13 × 1 = 13 = 1 × 13

(ii) 1 × 1 = 1 = 1 × 1

(iii) 25 × 1 = 25 = 1 × 25

(iv) 117 × 1 = 117 = 1 × 117

(v) 4295620 × 1 = 4295620

(vi) 108519 × 1 = 108519

We see that in every case a × 1 = a = 1 × a.

The #1 is known as the multiplication id or the id factor for multiplication of entire numbers as a result of it doesn’t change the id (worth) of the numbers through the operation of multiplication.

V. Associativity Property of Multiplication of Complete Numbers:

We are able to multiply three or extra numbers in any order. The

product stays the identical.

If a, b, c are any entire numbers, then

(a × b) × c = a × (b × c)

In different phrases, the multiplication of entire numbers is associative, that’s, the product of three entire numbers doesn’t change by altering their preparations.

When three or extra numbers are

multiplied, the product stays the identical no matter their group or place. We

can multiply three or extra numbers in any order, the product stays the identical.

For instance:

(i) (6 × 5) × 3 = 90

(ii) 6 × (5 × 3) = 90

(iii) (6 × 3) × 5 = 90

**Verification:**

With a view to confirm this property, we take three entire numbers say a, b, c and discover the values of the expression (a × b) × c and a × (b × c) as proven under :

**For instance:**

(i) (2 × 3) × 5 = 6 × 5 = 30 and a pair of × (3 × 5) = 2 × 15 = 30

Subsequently, (2 × 3) × 5 = 2 × (3 × 5)

(ii) (1 × 5) × 2 = 5 × 2 = 10 and 1 × (5 × 2) = 1 × 10 = 10

Subsequently, (1 × 5) × 2 = 1 × (5 × 2)

(iii) (2 × 11) × 3 = 22 × 3 = 66 and a pair of × (11 × 3) = 2 × 33 = 66

Subsequently, (2 × 11) × 3 = 2 × (11 × 3).

(iv) (4 × 1) × 3 = 4 × 3 = 12 and 4 × (1 × 3) = 4 × 3 = 12

Subsequently, (4 × 1) × 3 = 4 × (1 × 3).

(v) (1462 × 1250) × 421 = 1462 × (1250 × 421) = (1462 × 421)

× 1250

(vi) (7902 × 810) × 1725 = 7902 × (810 × 1725) = (7902 ×

1725) × 810

We discover that in every case (a × b) × c = a × (b × c).

Thus, the multiplication of entire numbers is associative.

VI. Distributive Property of Multiplication

of Complete Numbers / Distributivity of Multiplication over Addition of Complete Numbers:

When multiplier is the sum of two or extra numbers the

product is the same as the sum of merchandise.

If a, b, c are any three entire numbers, then

(i) a × (b + c) = a × b + a × c

(ii) (b + c) × a = b × a + c × a

In different phrases, the multiplication of entire numbers distributes over their addition.

**Verification:**

With a view to confirm this property, we take any three entire numbers a, b, c and discover the values of the expressions a × (b + c) and a × b + a × c as proven under :

**For instance:**

(i) 3 × (2 + 5) = 3 × 7 = 21 and three × 2 + 3 × 5 = 6 + 15 =21

Subsequently, 3 × (2 + 5) = 3 × 2 + 3 × 5

(ii) 1 × (5 + 9) = 1 × 14 = 15 and 1 × 5 + 1 × 9 = 5 + 9 = 14

Subsequently, 1 × (5 + 9) = 1 × 5 + 1 × 9.

(iii) 2 × (7 + 15) = 2 × 22 = 44 and a pair of × 7 + 2 × 15 = 14 + 30 = 44.

Subsequently, 2 × (7 + 15) = 2 × 7 + 2 × 15.

(vi) 50 × (325 + 175) = 50 × 3250 + 50 × 175

(v) 1007 × (310 + 798) = 1007 × 310 + 1007 × 798

These are the essential properties of multiplication of entire numbers.

Questions and Solutions on Properties of Multiplication:

**1. Fill within the Blanks.**

(i) Quantity × 0 = __________

(ii) 54 × __________ = 54000

(iii) Quantity × __________ = Quantity itself

(iv) 8 × (5 × 7) = (8 × 5) × __________

(v) 7 × _________ = 9 × 7

(vi) 5 × 6 × 12 = 12 × __________

(vii) 62 × 10 = __________

(viii) 6 × 32 × 100 = 6 × 100 × __________

**Solutions:**

(i) 0

(ii) 1000

(iii) 1

(iv) 7

(v) 79

(vi) 5 × 6

(vii) 620

(viii) 32

**2. Fill within the blanks utilizing Properties of Multiplication:**

(i) 62 × ………… = 5 × 62

(ii) 31 × ………… = 0

(iii) ………… × 9 = 332 × 9

(iv) 134 × 1 = …………

(v) 26 × 16 × 78 = 26 × ………… × 16

(vi) 43 × 34 = 34 × …………

(vii) 540 × 0 = …………

(viii) 29 × 4 × ………… = 4 × 15 × 29

(ix) 47 × ………… = 47

**Reply:**

**2. **(i) 5

(ii) 0

(iii) 332

(iv) 134

(v) 78

(vi) 43

(vii) 0

(viii) 15

(ix)

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