 # Playing with numbers Class 6 notes - Chapter 3

A number is defined as an arithmetical value, expressed by a word, symbol, and figures. These numbers can be written in single digits, double digits, three-digits in the generalized form.

## Types of Numbers

A number system is a system of writing for expressing numbers. According to the number system, the different types of a number includes:

• Prime numbers
• Even numbers
• Odd numbers
• Whole numbers
• Natural numbers
• Composite numbers To know more about Different Types of Numbers, visit here.
Lets us look into some solved example problems.

Write all the factors of 65
65 is a composite number.
65 = 1 × 65
5 x 13 = 65
Factors of 65: 1, 5, 13, 65.
Find the common factors of: 850 and 680
The common factors of 850 and 680 are 2, 5 and 17.

## Facts About Factors and Multiples

### Factors

A factor of a number is an exact divisor of that number.

Example: 1, 2, 3, and 6 are the factors of 6.

#### For More Information On Factors and Common Factors, Watch The Below Video. ### Properties of factors

Properties of factors of a number:

• 1 is a factor of every number.
• Every number is a factor of itself.
• Every factor of a number is an exact divisor of that number.
• Every factor is less than or equal to the given number.
• Number of factors of a given number are finite.

To know more about Factors and Mutiples, visit here.

### Perfect numbers

A number for which sum of all its factors is equal to twice the number is called a perfect number.

Example: Factors of 28 are 1, 2, 4, 7, 14 and 28.

Here, 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28
Therefore, sum of factors of 28 is equal to twice the number 28.
To know more about Perfect Numbers, visit here.

### Multiples

Multiples of a number are those numbers which we get on multiplying the number by any integer.
Example: Multiples of 3 are 6, 9, 12, 15, 18 etc.

### Properties of multiples

Properties of multiples of a number:

• Every multiple of a number is greater than or equal to that number.
• Number of multiples of a given number is infinite.
• Every number is a multiple of itself.

## Ones with the One and the Others

### Prime numbers

Numbers other than 1 whose only factors are 1 and the number itself are called Prime numbers.

Example: 2, 3, 5, 7 etc. To know more about Prime Numbers, visit here.

### Composite numbers

Numbers having more than two factors are called Composite numbers.

Example: 4, 6, 8 etc.
To know more about Composite Numbers, visit here.

## Divisible by 2 or 5 or Both

### Divisibility Tests

A divisibility rule is a method of determining whether a given integer is divisible by a fixed divisor without performing division, usually by examining its digits.
We have divisibility rules for 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11.

### Divisibility tests for 2

If one’s digit of a number is 0,2,4,6 or 8, then the number is divisible by 2.
Example: 12, 34, 56 and 78.

### Divisibility tests for 4

A number with 3 or more digits is divisible by 4 if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.
Example: 1396 is divisible by 4 since its last two digits i.e. 36 is divisible by 4.

### Divisibility tests for 3

A number is divisible by 3, if sum of its digits is divisible by 3.
Example: Take 27.
Sum of its digits = 2+7= 9, which is divisible by 3.
Therefore, 27 is divisible by 3. ### Divisibility tests for 5

If the one’s digit of a number is either 5 or 0, then it is divisible by 5.
Example: 75, 90, 100 and 125.

### Divisibility tests for 8

A number with 4 or more digits is divisible by 8, if the number formed by its last three digits is divisible by 8.
Example: 73512 is divisible by 8 since its last three digits i.e. 512 is divisible by 8.

### Divisibility tests for 6

If a number is divisible by 2 and 3 both, then it is divisible by 6 also.
Example: 120 is divisible by 2 and 3. Therefore, it is divisible by 6 also.

### Divisibility tests for 7

Double the last digit and subtract it from the remaining leading cut number. If result is divisible by 7, then the original number is divisible by 7. Example: 826 is divisible by 7 since, 82 – (6 × 2) = 82 – 12 =70, which is divisible by 7.

### Divisibility tests for 9

A number is divisible by 9 if sum of its digits is divisible by 9.
Example: Consider 126.
Sum of its digits = 1+2+6 = 9, which is divisible by 9.
Therefore, 126 is divisible by 9.

### Divisibility tests for 11

Find difference between sum of digits at odd places (from the right) and sum of digits at even places (from the right) of a number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.
Example: 1234321 is divisible by 11 since, (1+3+3+1) – (2+4+2) = 8 – 8 = 0, which is divisible by 11.

### Divisibility tests for 10

If one’s digit of a number is 0, then the number is divisible by 10.
Example: 10, 20, 30 and 40.
To know more about Divisibility Rules, visit here.

### Common factors

The factors of 4 are 1, 2 and 4.
The factors of 18 are 1, 2, 3, 6, 9 and 18.
The numbers 1 and 2 are common factors of both 4 and 18. ### Common multiples

Multiples of 3 are 3, 6, 9, 12, 15, 18,….
Multiples of 5 are 5, 10, 15, 20, 25, 30,…
Multiples of 6 are 6, 12, 18, 24, 30, 36,…
Therefore, common multiples of 3, 5 and 6 are 30, 60,….

## The Prime Factor

### Prime Factorization

When a number is expressed as a product of prime numbers, factorisation is called prime factorisation.

Example: Prime factorisation of 36 is 2×2×3×3.
To know more about Prime Factors, visit here.