 # Playing With Numbers Class 6 Notes Maths Chapter 3

## CBSE Class 6 Maths Notes Chapter 3 Playing With Numbers

A number which divides a given number are 1,2,3 and 6; exact divisors or factors of 15 are 1, exactly is called an exact divisor or factor of that 3, 5 and 15. number.Playing With Numbers Class 6 Notes Maths Chapter 3
For example exact divisors or factors of 6 are 1, 2, 3 and 6; exact divisors or factors of 15 are 1, 3, 5 and 15.

The different topics covered in CBSE Class 6 Mathematics Chapter 3 are tabulated below:

### Ex : 3.1 –  Introduction

• 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.
• A number which divides a given number are 1,2,3 and 6; exact divisors or factors of 15 are 1, exactly is called an exact divisor or factor of that 3, 5 and 15. number.
• For example exact divisors or factors of 6 are 1, 2, 3 and 6; exact divisors or factors of 15 are 1, 3, 5 and 15.

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

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.

### Ex : 3.2 –  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.

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.

Perfect numbers

A number for which the 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.

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.

### Ex : 3.3 –  Prime and Composite Numbers

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.

Composite numbers

Numbers having more than two factors are called Composite numbers.

Example: 4, 6, 8 etc.

### Ex : 3.4 –  Tests for Divisibility of Numbers

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 the 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 the 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 the 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.

### Ex : 3.5 –  Common Factors and Common Multiples

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,….

### Ex : 3.6 –  Some More Divisibility Rules

• If a number is divisible by another number, then it is also divisible by each of the factors of that number.
• For example, 40 is divisible by 20.
• Factors of 20 are 1, 2, 4, 5, 10 and 20.
• Clearly, 40 is divisible by each of these factors.
• If a number is divisible by two co-prime numbers, then it is also divisible by their product.
• For example, 40 is divisible by 4 and 5. 4 and 5 are co-prime.
• Their product is 4 × 5 = 20.
• Clearly, 40 is divisible by 20.
• If two given numbers are divisible by a number, then their sum is also divisible by that number.
• For example, The numbers 10 and 25 are divisible by a number 5.
• Their sum is 10 + 25 = 35.
• Clearly, 35 is divisible by 5.
• If two given numbers are divisible by a number, then their difference is also divisible by that number.
• For example, The numbers 10 and 25 are divisible by a number 5.
• Their difference is 25 – 10 = 15.
• Clearly, 15 is divisible by 5.

### Ex : 3.7 – Prime Factorisation

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.

### Ex : 3.8 – Highest Common Factor (HCF)

• The Highest Common Factor (HCF) of two or more given numbers is the highest (or greatest) of their common factors. It is also known as the Greatest Common Divisor (GCD).
• For example: Consider two numbers 12 and 20. Factors of 12 are 1, 2, 3, 4, 6 and 12. Factors of 20 are 1, 2, 4, 5, 10 and 20. The common factors of 12 and 20 are 1, 2 and 4. The highest of these is 4. So, 4 is the HGF of 12 and 20.

### Ex : 3.9 – Lowest Common Multiple (LCM)

• The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples.
• For example: Consider two numbers 12 and 20.
• Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, …
• Multiples of 20 are 20, 40, 60, 80, 100, 120, ……..
• The common multiples of 12 and 20 are 60, 120,…
• The lowest of these is 60.
• So, 60 is the LCM of 12 and 20.

### Ex : 3.10 – Some Problems on HCF and LCM

• In our everyday life, we are confronted with many situations in which we find it desirable to use the concepts of HCF and LCM.

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