**Multiplication of Integers**

We know that
multiplication is nothing but repeated addition.

**Example:**3 + 3 + 3 + 3 + 3 + 3 + 3 = 7 × 3 = 21

Similarly, (–3) + (–3) + (–3) + (–3) + (–3) + (–3) +
(–3) = 7 × (–3) = –21

Rules for multiplication of two integers are given below:

1. To find the
product of two integers with the same sign, find the product of the absolute
values of the two integers and assign plus sign to the product.

2. To find the
product of two integers having opposite signs, find the product of the absolute
values of the two integers and assign negative sign to the product.

For example,

a. 5 × 8 = 40, (–5) × (–8) = 40

b. 3 × (–8) = –24, (–3) × 8 = –24

**Example:**Find the product of the following integers.

a. 4 × 11 b. 3 × (–13) c. (–6) × 15 d. (–8) × (–7)

**Solution:**

a. 4 × 11 = 44

b. 3 × (–13) = – (3 × 13)
= –39

c. (–6) × 15 = – (6 × 15) = –90

d. (–8) × (–7) = 8 × 7 = 56

**Properties of Multiplication of
Integers**

###
**Closure
property**

If a and b are any two integers, then

*a**×***is also an integer.***b*
For example, 4 × 2 = 8 and 5 × (–6) = –30, Here,
8 and (–30) are also integers.

Thus, integers are closed for multiplication.

###
**Commutative
property**

If a and b are any two integers, then

*a**×**b = b**×**a*
For example, 5 × (–6) = (–6) × 5 = –30

Thus, integers are commutative for multiplication.

###
**Associative
property**

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

*(a**×**b)**× c**= a**×**(b**× c)*
For example, [2 × 5]
× (–6) = 2 × [5 × (–6)] = –60

Thus, integers are associative for multiplication.

###
**Distributive
property**

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

*a***× (***b***+***c***) =***a***×***b***+***a***×***c***and***a***× (***b***–***c***) =***a***×***b***–***a***×***c*
For example, a. 2 × (5
+ 6) = 2 × 5 + 2 × 6 = 10 + 12 = 22

b. 2 × (5
– 6) = 2 × 5 – 2 × 6 = 10 – 12 = –2

Thus, integers are distributive for multiplication
over addition and subtraction

###
**Multiplicative
identity**

If a is
an integer, then

**a****×****1 = 1****×****a = a.**
Here, 1
is called the multiplicative identity.

###
**Multiplicative
property of zero**

If a is
an integer, then

**a****×****0 = 0****×****a = 0.**
Thus, if
0 is multiplied with any integer, the product is always 0.

**Example 1:**Take three integers –7, 8 and –5 and check if (

*a*×

*b*) ×

*c*=

*a*× (

*b*×

*c*)?

**Solution:**

L.H.S. = (

*a*×*b*) ×*c*= [(–7) × 8] × (–5) = (–56) × (–5) = 280
R.H.S. =

*a*× (*b*×*c*) = (–7) × [8 × (–5)] = (–7) × (–40) = 280
Hence, (

*a*×*b*) ×*c*=*a*× (*b*×*c*)**Example 2:**Evaluate the following using properties of multiplication.

a. 2 × (–6) × (–3) b. 7 × 97 c. 12 × 104

**Solution:**

a. 2 × (–6) × (–3) = [2 ×
(–6)] × (–3) (Using associative property)

= (–12) × (–3) = 36

b. 7 × 97 = 7 × (100 – 3)
= 7 × 100 – 7 × 3 (Using distributive property)

= 700 – 21 = 679

c. 12 × 104 = 12 × (100 +
4) = 12 × 100 + 12 × 4 (Using distributive property)

= 1200 + 48 = 1248

##
**Division of Integers **

**Division**is the inverse of multiplication. The rules for the division of integers are as follow:

1. Quotient of two positive integers is a positive
integer.

2. Quotient of two negative integers is a positive
integer.

3. Quotient of two integers with
different signs is a negative integer.

Dividing
45 by –9 means finding an integer which when multiplied by –9, gives 45. The
required integer is –5. Therefore, we write 45 ÷ (–9) = –5.

**Example:**Find the value of the following:

a. (–18) ÷ (9) b. (–22) ÷ (–2) c. 56 ÷ (–4) d. (–215) ÷ (–5)

**Solution:**

a. (–18) ÷ (9) = –18/9
= –2

b. (–22) ÷ (–2) = –22/–2 = 11

c. 56 ÷ (–4) = 56/–4
= –14

d. (–215) ÷ (–5) = --215/–5
= 43

##
**Properties of Division of Integers**

###
**Closure
property**

If a and b are any two integers, then

*a***÷****is not always an integer.***b*
For example, 16 ÷ 2 = 8 but
15 × (–6) = –2.5, Here, 2.5 is not an integer.

Thus, integers are not closed for division.

###
**Commutative
property**

If a and b are any two integers, then

*a***÷***b ≠ b***÷***a*
For example, 12 ÷
(–6)

*≠*(–6) ÷ 12
Thus, integers are not commutative for division.

###
**Associative
property**

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

*(a***÷***b)***÷***c**≠ a***÷***(b***÷***c)*
For example, [15 ÷ 5]
÷ (–3) = 15 ÷
[5 ÷ (–3)]

Thus, integers are not associative for division.

###
**Division of an integer
by itself and by 1**

If
a is an integer, then

*a***÷***a***= 1**, where*a*≠ 0 and*a***÷ 1 =***a*###
**Division property of zero**

If
a is an integer
and

*a*≠ 0, then**0 ÷***a***= 0**###
**Division of an integer by its
additive inverse and –1**

If a is an integer, then

*a***÷ (–***a***) = –1, (–***a***) ÷***a***= –1,***a***÷ (–1) = –***a*

##
**Word
Problems on Integers**

**Example 1:**A snail climbs 21 cm up a wall and falls back 11 cm in one hour. How much will it climb in 6 hours?

**Solution:**The snail climbs +21 cm. Falls back –11 cm.

In 6 hours, the distance it will cover = 6[21 + (–11)]
cm = 6(21 – 11) = 6 × 10 cm = 60 cm.

**Example 2:**The temperature of an AC room was 24 °C at 9 p.m. The temperature dropped by 3 °C in one hour. What is the temperature at 10 p.m.?

**Solution:**Temperature at 9 p.m. = 24 °C and change in temperature = –3 °C

Thus, temperature at 10 p.m. = 24 °C – 3 °C = 21 °C

##
**Simplification
of Integers using BODMAS**

To
simplify an expression with two or more basic fundamental operations, grouping
is used. The grouping symbols are called

**brackets**. There are 4 types of brackets.
1. — is called

**bar**or**vinculum**.
2. ( ) are called

**parentheses**.
3. { } are called

**braces**or**curly brackets**.
4. [ ] are called

**square brackets**.
The brackets are removed in the
following order.

a. bar (—)

b. parentheses ( )

c. braces { }

d. square brackets [ ]

To simplify problems containing
brackets, the order of operation is remembered by the word

**BODMAS**where the letters have the meaning as shown below:
B → Brackets

O → Of

D → Division

M → Multiplication

A → Addition

S → Subtraction

**Example:**Simplify the following.

a. 35 + 20 ÷ {3 – (2
+ 5)} b. 5 – [6 + {19 – (16 – 2)}].

**Solution:**

a. 35
+ 20 ÷ {3 – (2 + 5)} = 35 + 20 ÷ {3 – 7}

= 35 + 20 ÷ (–4) = 35 – 5 =
30

b.
15 – [6 + {19 –
(16 – 2)}] = 15 – [6 + {19 – 14}] (Using BODMAS)

= 15 – [6 + 5] = 15 – 11 = 4

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