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Playing with Numbers Class 6 Chapter 3 Maths Notes
Numbers are arithmetic values expressed by words, symbols, and figures. In general, these numbers can be written as single digits, double digits, or three digits.
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 include:
- Prime numbers
- Even numbers
- Odd numbers
- Whole numbers
- Natural numbers
- Composite numbers
Let’s look at some solved examples:
- 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.
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 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 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.
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.
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 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 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 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 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 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 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 10
If one’s digit of a number is 0, then the number is divisible by 10.
Example: 10, 20, 30 and 40.
- Divisibility tests for 11
Find the difference between the 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.
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.
Also Read –
- Knowing our Numbers: Class 6 Chapter 1 Maths Notes
- Whole Numbers: Class 6 Chapter 2 Maths Notes
- Food: Where does it come from? Class 6 Notes
- Components of Food: Class 6 Chapter 2 Science Notes
- Fibre to Fabric: Class 6 Chapter 3 Science Notes
- Sorting Materials into Groups Class 6 Chapter 4 Science Notes
- Separation of Substances: Class 6 Chapter 5 Science Notes
- Changes Around Us: Class 6 Chapter 6 Science Notes
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