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

A number that 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. *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 that 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.
- 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.
- A number of factors of a given number is 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 that we get on multiplying the number by an 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 the sum of digits at odd places (from the right) and the 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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