Properties Of Multiplication Of Integers

1. Closure under Multiplication

In case of multiplication, the product of two integers is always integer so integers are closed under multiplication.

For all the integers p and q

p \times q = r , where r is an integer

Example

(-10) \times (-3) = 30

(12) \times (-4) = -48

2. Commutativity of Multiplication

If we change the order of the integers while multiplying then also the result will remain the same then it is said that multiplication is commutative for integers.

For any two integers p and q

p \times q = q \times p

Example

20 \times (-30) = - 600

(-30) \times 20 = - 600

3. Multiplication by Zero

If we multiply an integer with zero then the result will always be zero.

For any integer p,

p \times 0 = 0 \times p = 0

Example

9 \times 0 = 0 \times 9 = 0

0 \times \left(-15 \right) = 0

4. Multiplicative Identity

If we multiply an integer with 1 then the result will always the same as the integer.

For any integer q

q \times 1 = 1 \times q = q

Example

21 \times 1 = 1 \times 21 = 21

1 \times (-15) = (-15)

5. Associative Property

If we change the grouping of the integers while multiplying in case of more than two integers and the result remains the same then it is said the associative property for multiplication of integers.

For any three integers, p, q and r

p \times (q \times r ) = (p \times q ) \times r

6. Distributive Property

a. Distributivity of Multiplication over Addition.

For any integers a, b and c

a \times (b + c ) = (a \times b ) + (a \times c )

Example

Solve the following by distributive property.

I. 35 \times (10 + 2) = 35 \times 10 + 35 \times 2

= 350 + 70

= 420

II. (- 4) \times [(-2) + 7] = (- 4) \times 5 = - 20 And

=[(- 4) \times (-2)] +[(- 4) \times 7]

= 8 + (-28)

=-20

So, (- 4) \times [(-2) + 7] = [(- 4) \times (-2)] + [(- 4) \times 7]

b. Distributivity of multiplication over subtraction

For any integers a, b and c

a \times (b - c ) = (a \times b ) - (a \times c )

Example

5 \times (3 - 8) = 5 \times (- 5) = - 25

5 \times 3 - 5 \times 8 = 15 - 40 = - 25

So, 4 \times (3 - 8) = 4 \times 3 - 4 \times 8

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