# Multiplying and Dividing Integers

Multiplying and dividing integers is similar to what we know in whole numbers. To multiply and divide integers with positive and negative signs, we multiply their absolute values and then follow the rules given below:

## Multiplying Integers

There can be 2 possible situations, and based on them, the rules are given below:

### With the Same Sign

When multiplying integers having the same sign, the product is always a positive integer. We multiply the absolute values and give a positive sign before the product.

Multiplying a positive integer with a positive integer gives a positive integer.

Positive × Positive = Positive, for example, 4 × 6 => |4| × |6| = 24

Similarly, multiplying a negative integer with a negative integer also gives a positive integer.

Negative × Negative = Positive, for example, (-4) × (-6) = +(|-4| × |-6|) => 4 × 6 = 24

### With Different Signs

The result of multiplying integers with different signs (+and -) is always a negative integer. We multiply the absolute values and give a negative sign before the product.

Multiplying a positive and a negative integer gives a negative integer.

Positive × Negative = Negative, for example, 4 × (-6) = -(|4| × |-6|) => -(4 × 6)  => -24

Multiplying a negative integer with a positive integer also gives a negative integer.

Negative × Positive = Negative, for example, (-4) × 6 => -(|-4| × |6|) => -24

### Solved Example

Mutiply:
a) 3 × (-12)
b) (-8) × (-4)
c) (-3) × 11
d) 4 × 12

Solution:

a)  3 × (-12) => -(|3| × |-12|) => -36
b) (-8) × (-4) => +(|-8| × |-4|) => 32
c) (-3) × 11 => -(|-3| × |11|) => -33
d) 4 × 12 => |4| × |12| => 48

## Dividing Integers

Similar to multiplication, there can be 2 possible situations. Based on them, the rules are given below:

### With the Same Sign

When dividing integers having the same sign, the quotient is always positive. Here, we divide the absolute values and give a positive sign before the quotient.

Dividing a positive integer with a positive integer gives a positive integer.

Positive ÷ Positive = Positive, for example, (+24) ÷ (+6) => |24| ÷ |6| => 4

Similarly, dividing a negative integer by a negative integer also gives a positive integer.

Negative ÷ Negative = Positive, for example, (-24) ÷ (-6) => +(|-24| ÷ |-6|) => 4

### With Different Signs

When dividing integers with different signs, the result is always negative. Here, we divide the absolute values and give a negative sign before the quotient.

Dividing a positive integer with a negative integer gives a negative integer.

Positive ÷ Negative = Negative, for example, (+24) ÷ (-6) => -(|24| ÷ |-6|) => -4

Dividing a negative integer with a positive integer also gives a negative integer.

Negative ÷ Positive = Negative, for example, (-24) ÷ (+6) => -(|-24| ÷ |6|) => -4

### Solved Example

Divide:
a) (+36) ÷ (-2)
b) (-44) ÷ (-4)
c) (+27) ÷ (+9)
d) (-54) ÷ (+8)

Solution:

a) (+36) ÷ (-2) => -(|36| ÷ |-2|) => -18
b) (-44) ÷ (-4) => +(|-44| ÷ |-4|) => 11
c) (+27) ÷ (+9) => |27| ÷ |9| => 3
d) (-54) ÷ (+8) => -(|-54| ÷ |8|)  => -6

## Properties

1. Closure Property: It states that when an integer is multiplied by another integer, the product is also an integer. However, it is only sometimes valid for division. Thus, multiplication is closed for integers.

For example,

2 × 5 = 10 is also an integer

2 ÷ 5 = 2/5 is not an integer; however 6 ÷ 2 = 3 is an integer

2. Commutative Property: It states that interchanging the positions of operands in multiplication does not change the result. However, it is not true for division.

For example,

(-2) × 5 = 5 × (-2) = -10

(-2) ÷ 5 ≠ 5 ÷ (-2)

3. Associative Property: It states that changing the grouping of integers does not change the result during multiplication. However, it is not true for division.

For example,

((-2) × 5) × 3 = (-2) × (5 × 3) = -30

((-2) ÷ 5) ÷ 3 ≠ (-2) ÷ (5 ÷ 3)

4. Distributive Property: It states that an expression of form a (b + c) can be solved as a × b + a × c in the case of multiplication. However, it is not true for division.

For example,

5 × (-3 + 6) = 5 × (-3) + 5 × (6) = 15

5 ÷ (-3 + 6) ≠ 5 ÷ (-3) + 5 ÷ (6)

5. Identity Property: It states that for any integer ‘a’ for multiplication, a × 1 = 1 × a. However, there is no such identity element in the division of integers.

For example,

5 × 1 = 1 × 5 = 5