Page 38 - Start Up Mathematics_6

III. Multiplication
 Closure property
Let ‘a’ and ‘b’ be any two whole numbers, then their product ‘a × b’ is also a whole number.
This property is known as closure property of multiplication of whole numbers.

For example, 3 × 6 = 18

 Commutative property
Let ‘a’ and ‘b’ be any two whole numbers, then a × b = b × a.
The product of whole numbers is commutative.

For example, 3 × 5 = 15 = 5 × 3

 Existence of multiplicative identity
Let ‘a’ be any whole number, then a × 1 = a = 1 × a.

The number 1 is called the multiplicative identity for whole numbers.
For example, 7 × 1 = 7 = 1 × 7

 Associative property
Let ‘a’, ‘b’ and ‘c’ be any three whole numbers, then (a × b) × c = a × (b × c).

The product of numbers remains unchanged if the grouping is changed.
For example, (2 × 3) × 4 = 24 = 2 × (3 × 4)

 Property of zero
Let ‘a’ be any whole number, then a × 0 = 0 = 0 × a.

For example, 4 × 0 = 0 = 0 × 4
 Distributive property of multiplication over addition

Let ‘a’, ‘b’ and ‘c’ be any three whole numbers, then a × (b + c) = a × b + a × c.
In whole numbers, multiplication distributes over addition.
For example, 4 × (5 + 7) = 4 × 5 + 4 × 7
LHS = 4 × (5 + 7) = 4 × 12 = 48

RHS = 4 × 5 + 4 × 7 = 20 + 28 = 48

• Distributive property can also be written as, (b + c) × a = b × a + c × a.
• Multiplication distributes over subtraction also, a × (b – c) = a × b – a × c, provided b ≥ c.

Example 6: Find the products by suitable rearrangement.
(a) 4 × 167 × 25 (b) 625 × 348 × 16 (c) 125 × 40 × 8 × 25
Solution: (a) 4 × 167 × 25 = (4 × 25) × 167 = 100 × 167 = 16,700

(b) 625 × 348 × 16 = (625 × 16) × 348 = 10,000 × 348 = 34,80,000
(c) 125 × 40 × 8 × 25 = (125 × 8) × (40 × 25) = 1,000 × 1,000 = 10,00,000

Example 7: Find the values of the following:
(a) 297 × 17 + 297 × 3 (b) 38,443 × 94 + 6 × 38,443
(c) 81,265 × 169 – 81,265 × 69

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