There are six integer multiplication properties that help solve the problems easily.

The six properties of multiplication are closure property, commutative property, zero property, identity property, associativity property and distributive property.

The properties of integer multiplication are discussed below; These properties will help us find the product convenient, even in very large numbers.

Closure property of whole numbers:

If a and b are two numbers, then their product a × b is also an integer.

In other words, if we multiply two integers, we get an integer.

**Verification:**

To check this property, we take a few pairs of integers and multiply them;

**For example:**

(i) 8 × 9 = 72

(ii) 0x16 = 0

(iii) 11 × 15 = 165

(iv) 20 × 1 = 20

We find that the product is always an integer.

Commutativity of whole numbers / order property of whole numbers:

The multiplication of whole numbers is commutative.

In other words, if a and b are any two integers, then a × b = b × a.

We can multiply numbers in any order. The product does not change when the number order is changed.

When multiplying any two numbers, the product stays the same regardless of the order of the multiplicands. We can multiply numbers in any order, the product stays the same.

For example:

(i) 7 × 4 = 28

(ii) 4 × 7 = 28

**Verification:**

To check this property, let’s take some pairs of integers and multiply those numbers in different orders as shown below;

**For example:**

(i) 7 × 6 = 42 and 6 × 7 = 42

Hence 7 × 6 = 6 × 7

(ii) 20 × 10 = 200 and 10 × 20 = 200

Hence 20 × 10 = 10 × 20

(iii) 15 × 12 = 180 and 12 × 15 = 180

Hence 15 × 12 = 12 × 15

(iv) 12 × 13 = 156 and 13 × 12

Hence 12 × 13 = 13 × 12

(V) 1122 × 324 = 324 × 1122

(vi) 21892 × 1582 = 1582 × 21892

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

III. Multiplication by zero / zero Property of multiplying whole numbers:

When a number is multiplied by 0, the product is always 0.

If a is an integer, then a × 0 = 0 × a = 0.

In other words, the product of an integer and zero is always zero.

If 0 is multiplied by any number, the product is always zero.

For example:

(i) 3 × 0 = 0 + 0 + 0 = 0

(ii) 9 × 0 = 0 + 0 + 0 = 0

**Verification:**

To check this property we take some whole numbers and multiply them by zero as shown below;

**For example:**

(i) 20 × 0 = 0 × 20 = 0

(ii) 1 × 0 = 0 × 1 = 0

(iii) 115 × 0 = 0 × 115 = 0

(iv) 0x0 = 0x0 = 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 an integer and zero is zero.

IVMultiplicative identity of whole numbers / identity property of whole numbers:

When a number is multiplied by 1, the product is the number itself.

If a is an integer, then a × 1 = a = 1 × a.

In other words, the product of an integer and 1 is the number itself.

When 1 is multiplied by any number, the product is always the number itself.

For example:

(i) 1 × 2 = 1 + 1 = 2

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

**Verification:**

To verify this property, we find the product of various integers with 1 as shown 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 a × 1 = a = 1 × a in each case.

The number 1 is called the multiplication identity or integer multiplication identity element because it does not change the identity (value) of the numbers during the multiplication process.

Associativity property of integer multiplication:

We can multiply three or more numbers in any order. The product stays the same.

If a, b, c are any integers, then

(a × b) × c = a × (b × c)

In other words, whole number multiplication is associative, meaning that the product of three whole numbers does not change by changing their order.

When three or more numbers are multiplied, the product will stay the same regardless of their group or place. We can multiply three or more numbers in any order, the product stays the same.

For example:

(i) (6 × 5) × 3 = 90

(ii) 6 × (5 × 3) = 90

(iii) (6 × 3) × 5 = 90

**Verification:**

To check this property, let’s take three integers, say a, b, c, and find the values of the expression (a × b) × c and a × (b × c) as shown below:

**For example:**

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

Hence (2 × 3) × 5 = 2 × (3 × 5)

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

Hence (1 × 5) × 2 = 1 × (5 × 2)

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

Hence (2 × 11) × 3 = 2 × (11 × 3).

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

Hence (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 find that in each case (a × b) × c = a × (b × c).

The multiplication of whole numbers is therefore associative.

Distribution property of the multiplication of whole numbers / distributivity of the multiplication over the addition of whole numbers:

If the multiplier is the sum of two or more numbers, the product is equal to the sum of the products.

If a, b, c are any three integers, then

(i) a × (b + c) = a × b + a × c

(ii) (b + c) × a = b × a + c × a

In other words, the multiplication of integers is divided into their addition.

**Verification:**

To check this property, we take any three integers a, b, c and find the values of the expressions a × (b + c) and a × b + a × c as shown below:

**For example:**

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

Hence 3 × (2 + 5) = 3 × 2 + 3 × 5

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

Hence 1 × (5 + 9) = 1 × 5 + 1 × 9.

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

Hence 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 important properties of integer multiplication.

Questions and answers on properties of multiplication:

**1. Fill in the gaps.**

(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 × __________

**Reply:**

(i) 0

(ii) 1000

(iii) 1

(iv) 7

(v) 79

(vi) 5 × 6

(vii) 620

(viii) 32

**2. Fill in the gaps with 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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