**Properties of multiplication of integers :**

In Math, the whole numbers and negative numbers together are called integers. The set of all integers is denoted by Z.

Z = {... - 2, - 1,0,1,2, ...}, is the set of all integers

Here, we are going to see the following three properties of multiplication of integers.

(i) Closure property

(ii) Commutative property

(iii) Associative property

Let us look at these properties one by one in detail.

Observe the following:

– 10 × (– 5) = 50

40 × (– 15) = – 600

In general, a × b is an integer, for all integers a and b.

Therefore, integers are closed under multiplication.

Observe the following :

5 × (– 6) = – 30 and (– 6) × 5 = – 30

Therefore, 5 × (– 6) = (– 6) × 5

Therefore, multiplication is commutative for integers.

In general, for any two integers a and b, a × b = b × a.

The product of any nonzero integer with zero is zero.

Observe the following:

5 × 0 = 0

– 8 × 0 = 0

In general, for any nonzero integer a

a × 0 = 0 × a = 0

Observe the following:

5 x 1 = 5

1 x (- 7) = -7

This shows that ‘1’ is the multiplicative identity for integers.

In general, for any integer a we have

a x 1 = 1 x a = a

Consider the integers 2, – 5, 6.

Look at

[2 x (-5)] x 6 = -10 x 6 = -60 and

2 x [(- 5) x 6] = 2 x (-30) = -60

Thus [2 x (-5)] x 6 = 2 x [(- 5) x 6]

So we can say that integers are associative under multiplication.

In general, for any integers a, b, c, (a × b) × c = a × (b × c).

**Multiplication is distributive over addition :**

Consider the integers 12, 9, 7.

Look at

12 x (9 + 7) = 12 x 16 = 192

12 x (9 + 7) = 12 x 9 + 12 x 7 = 108 + 84 = 192

Thus 12 x (9 + 7) = (12 x 9) + (12 x 7)

In general, for any integers a, b, c.

a x (b + c) = (a x b) + (a x c).

Therefore, multiplication is distributive over addition of integers.

**Multiplication is distributive over subtraction :**

Consider the integers 12, 9, 7.

Look at

12 x (9 - 7) = 12 x 2 = 24

12 x (9 - 7) = 12 x 9 - 12 x 7 = 108 - 84 = 24

Thus 12 x (9 - 7) = (12 x 9) - (12 x 7)

In general, for any integers a, b, c.

a x (b - c) = (a x b) - (a x c).

Therefore, multiplication is distributive over addition of integers.

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