Counting Formula and Divisibility Rules (Basic and Advanced) - Brinity Institute (Delhi)

                                                                                               
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Some Rules on Counting Numbers :

Divisibility Rules (Basic and Advanced )

Divisibility by 2 : A number is divisible by 2, if the last digit is even.

Divisibility by 3 : A number is divisible by 3, if the sum of its digit is divisible by 3.
for example, 91551, sum of digits = 9+1+5+5+1= 21, 21 is divisible by 3. So 91551 is also divisible by 3.

Divisibility by 4 : A number is divisible by 4, if the last two digits are divisible by 4.

Divisibility by 5 : A number is divisible by 5, if the last digit is either 0 or 5.

Divisibility by 6 : A number is divisible by 6, if it is simultaneously divisible by 2 and 3 both.

Divisibility by 8 : A number is divisible by 8, if the last three digits are divisible by 8.

Divisibility by 9 : A number is divisible by 9, if sum of its digit is divisible by 9.

Divisibility by 10 : A number is divisible by 10, if the last digit is 0.



Any digit repeated 6 times will be always divisible by 3,7,11,13,37
for example : 111111, 222222, 444444, 666666, etc. are always divisible by 3,7,11,13, 37.



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Numbers Around Us - Brinity Institute (Delhi)

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Classification of Numbers :

Natural Numbers : Numbers which we use for counting the objects are known as natural numbers. They are denoted by ‘N’ .
                     N = {1,2,3,4,5.......}

Whole Numbers : When we include ‘zero’ in the set of natural numbers , the set  is known as whole numbers . They are denoted by ‘W’
                    W = { 0,1,2,3,4,5,........}

Integers : All the natural numbers and their negatives including zero form the set of integers.

NOTE  : Zero is neither positive nor negative. It is neutral.
Fractions : Numbers of the form p/q (q≠0) such that p and q both are integers and the net result of  does not evaluate to an integer.
e.g.  3/7, 16/5, 3 + 2/5 etc.
NOTE :  6/2 is not a fraction as the result comes out to be 3.

Rational Numbers  :  Numbers that can be expressed as p/q , where p and q are integers and q≠0.
Rational numbers are expressible either in terminating decimals or in repeating decimals.

Irrational Numbers : The numbers which cannot be expressed in p/q form . Non terminating and non repeating decimals are called as irrational numbers.



Conversion of Recurring decimal to corresponding fractions :

 

Prime Numbers: The numbers  which have no divisors/factors apart from 1 and itself.


Properties of Prime Numbers  :





How to Check whether a given number is a prime or not.
To check whether a number N is prime or not, adopt the following steps:

Take the square root of the given number and round of the square root to immediately lower integer. Call this number ‘z’. For example if you have to check for 181, then its square root will lie between 13 and 14 ( 13 < z < 14 ) . Take z=13.

Check for the divisibility of the number N by all primes below z. If there is no prime number below the value of z which divides N then N will be Prime.

For e.g  Let N=239, square root of 239 lies between 15 and 16. Hence, take the value of z is 15. Prime numbers less than 15 are 2,3,5,7,11 and 13.  239 is not divisible by any of the primes. Hence we can conclude that 239 is a prime number.


Composite Number :   A number other than 1, which is not a prime number is called a composite number OR A number which has atleast three distinct factors (including 1) is called a composite number.


Perfect Numbers: IF the sum of all the factors of a number is equal to the twice of the number then that number is called perfect number. e.g. 6, 28, etc.

Co-Prime Numbers : Two numbers 'a' and 'b' are co-prime to each other iff HCF (a,b)=1.
e.g. (10,21)



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