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.
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The different topics covered in CBSE Class 6 Mathematics Chapter 3 are tabulated below:
| Exercise | Topics |
| 3.1 | Introduction |
| 3.2 | Factors and Multiples |
| 3.3 | Prime and Composite Numbers |
| 3.4 | Tests for Divisibility of Numbers |
| 3.5 | Common Factors and Common Multiples |
| 3.6 | Some More Divisibility Rules |
| 3.7 | Prime Factorisation |
| 3.8 | Highest Common Factor |
| 3.9 | Lowest Common Multiple |
| 3.10 | Some Problems on HCF and LCM |
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.
CBSE Notes for Class 6 Maths Free Download for All Chapters
| CBSE Class 6 Maths Study Notes | CBSE Class 6 Maths Study Notes |
| Knowing our Numbers Class 6 Notes Chapter 1 | Decimals Class 6 Notes Chapter 8 |
| Whole Numbers Class 6 Notes Chapter 2 | Data Handling Class 6 Notes Chapter 9 |
| Playing with Numbers Class 6 Notes Chapter 3 | Mensuration Class 6 Notes Chapter 10 |
| Basic Geometrical Ideas Class 6 Notes Chapter 4 | Algebra Class 6 Notes Chapter 11 |
| Understanding Elementary Shapes Class 6 Notes Chapter 5 | Ratio And Proportion Class 6 Notes Chapter 12 |
| Integers Class 6 Notes Chapter 6 | Symmetry Class 6 Notes Chapter 13 |
| Fractions Class 6 Notes Chapter 7 | Practical Geometry Class 6 Notes Chapter 14 |
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