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THE MAGIC OF NATURE POEM WRITTEN BY SAFALYA PAL

THE MAGIC OF NATURE THE MAGIC OF NATURE OUR WONDER DOES GROW THE SECRETS OF NATURE ARE NOT FOR THE HUMANS TO KNOW THE MAGIC OF NATURE IS FOR ALL TO SEE IT IS ALL AROUND YOU AND ALL AROUND ME THE BEUTIFUL SCENIC VIEW EVERYTHING SEEMS AWESOME AND NEW LOOKING AT THE AWESOME SIGHT YOU FEEL SO TRANQUIL AND DELIGHT THINK HOW THE LUSH FORESTS ALWAYS REMAIN GREEN THINK WHY THE WATER DOESN’T ACCOMPLISH THINK HOW THE NORTHERN LIGHTS ESTABLISH THINK FROM WHERE DO WE GET NATURAL THINGS WE GET ALL THESE FROM THE NATURE AND THOSE ARE POSSIBLE FOR NATURE’S MAGIC! THE NATURAL BEAUTY OF NATURE HAS SO MUCH TO SAY, THE BEAUTIFUL CROSSROADS AND LONG WAY, THE NATURE HAS ITS CHARM, KEEP IT SAFE AND DO NOT HARM, THE NATURE WILL KEEP US UNSAFE, IF WE DON'T KEEP IT SAFE!                                          ...
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Playing with Numbers. Class 6 NCERT NOTES

Playing with Numbers

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.

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 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.

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.

Composite numbers

Numbers having more than two factors are called Composite numbers.

Example: 4, 6, 8 etc.

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.

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.

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