There are 6 residential or commercial properties of reproduction of entire numbers that

will assist to resolve the issues quickly.

The 6 residential or commercial properties of reproduction are Closure Property, Commutative Property,

No Property, Identity Property,

Associativity Property and Distributive

Home.

The residential or commercial properties of reproduction on entire numbers are talked about listed below; these residential or commercial properties will assist us in discovering the item of even large numbers easily.

Closure Property of Whole Numbers:

If a and b are 2 numbers, then their item a × b is likewise an entire number.

In other words, if we increase 2 entire numbers, we get an entire number.

** Verification: **

In order to confirm this home, let us take a couple of sets of entire numbers and increase them;

** For example: **

( i) 8 × 9 = 72

( ii) 0 × 16 = 0

( iii) 11 × 15 = 165

( iv) 20 × 1 = 20

We discover that the item is constantly an entire numbers.

Commutativity of Whole Numbers/ Order Property of Whole Numbers:

The reproduction of entire numbers is commutative.

In other words, if a and b are any 2 entire numbers, then a × b = b × a.

We can increase numbers in any order. The item does not

When the order of numbers is altered, modification. When increasing,

any 2 numbers, the item stays exact same despite the order of

multiplicands. We can increase numbers in any order, the item stays the

exact same.

For Example:

( i) 7 × 4 = 28

**( ii) 4 × 7 = 28**

Verification:

** In order to confirm this home, let us take a couple of sets of entire numbers and increase these numbers in various orders as revealed listed below;**

For Example:

( i) 7 × 6 = 42 and 6 × 7 = 42

Therefore, 7 × 6 = 6 × 7

( ii) 20 × 10 = 200 and 10 × 20 = 200

Therefore, 20 × 10 = 10 × 20

( iii) 15 × 12 = 180 and 12 × 15 = 180

Therefore, 15 × 12 = 12 × 15

( iv) 12 × 13 = 156 and 13 × 12

Therefore, 12 × 13 = 13 × 12

( V) 1122 × 324 = 324 × 1122

( vi) 21892 × 1582 = 1582 × 21892

We discover that in whatever order we increase 2 entire numbers, the item stays the exact same.

III. Reproduction By Zero/Zero Property of Multiplication of Whole Numbers:

When a number is increased by 0, the item is constantly 0.

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

In other words, the item of any entire number and absolutely no is constantly absolutely no. When 0 is increased by any number the,

item is constantly absolutely no.

For example:

**( i) 3 × 0 = 0 + 0 + 0 = 0**

( ii) 9 × 0 = 0 + 0 + 0 = 0

** Verification: **

In order to confirm this home, we take some entire numbers and increase them by absolutely no as revealed listed below;

For example:

( 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 item of any entire number and absolutely no is absolutely no.

IV. Multiplicative Identity of Whole Numbers/ Identity Property of Whole Numbers:

When a number is increased by 1, the item is the number

itself.

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

In other words, the item of any entire number and 1 is the number itself. When 1 is increased by any number the,

item is constantly the number itself.

** For example: **

( i) 1 × 2 = 1 + 1 = 2

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

Verification:

In order to confirm this home, we discover the item of various entire numbers with 1 as revealed listed below:

For example:

( 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 each case a × 1 = a = 1 × a.

The number 1 is called the reproduction identity or the identity aspect for reproduction of entire numbers since it does not alter the identity (worth) of the numbers throughout the operation of reproduction.

V. Associativity Property of Multiplication of Whole Numbers:

We can increase 3 or more numbers in any order. The

item stays the exact same.

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

( a × b) × c = a × (b × c)

In other words, the reproduction of entire numbers is associative, that is, the item of 3 entire numbers does not alter by altering their plans. When 3 or more numbers are,

increased, the item stays the exact same despite their group or location. We

can increase 3 or more numbers in any order, the item stays the exact same.

** For example: **

( i) (6 × 5) × 3 = 90

**( ii) 6 × (5 × 3) = 90**

( iii) (6 × 3) × 5 = 90

Verification:

In order to confirm this home, we take 3 entire numbers state a, b, c and discover the worths of the expression (a × b) × c and a × (b × c) as revealed listed below:

For example:

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

Therefore, (2 × 3) × 5 = 2 × (3 × 5)

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

Therefore, (1 × 5) × 2 = 1 × (5 × 2)

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

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

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

Therefore, (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 each case (a × b) × c = a × (b × c).

Thus, the reproduction of entire numbers is associative.

VI. Distributive Property of Multiplication.

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

When multiplier is the amount of 2 or more numbers the.

item amounts to the amount of items.

** If a, b, c are any 3 entire numbers, then**

( i) a × (b + c) = a × b + a × c

**( ii) (b + c) × a = b × a + c × a**

In other words, the reproduction of entire numbers disperses over their addition.

Verification:

In order to confirm this home, we take any 3 entire numbers a, b, c and discover the worths of the expressions a × (b + c) and a × b + a × c as revealed listed below:

For example:

( 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.

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

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

These are the crucial residential or commercial properties of reproduction of entire numbers.

Questions and Answers on Properties of Multiplication:

1. Complete the Blanks.

( i) Number × 0 = __________

( ii) 54 × __________ = 54000

( iii) Number × __________ = Number 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 × __________

Answers:

( i) 0

( ii) 1000

( iii) 1

( iv) 7

**( v) 79**

( vi) 5 × 6

( vii) 620

( viii) 32

2. Complete 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

Answer:

2.

( i) 5

( ii) 0

( iii) 332

( iv) 134

( v) 78

**( vi) 43****Whole Numbers**

**Numbers Page****( vii) 0**

**6th Grade Page****( viii) 15**

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