Assorted Divisibilty Rules

Assorted Divisibility Rules :-

I ) Divisibility By 2 : A number is divisible by 2 if its unit is any of 0 ,2 ,4 , 6 , 8 .


Ex. 86354 is divisible by 2 , while 673453 is not divisible by 2.


II) Divisibility By 3 : A number is divisible by 3 only when the Sum of its digits is


divisible by 3 .


Ex. a) In the number 765432 ,the Sum of its digits = 27 , which is divisible by 3 .


Hence 765432 is divisible by 3 .


b) In the number 489365 ,the Sum of its = 35 ,which is not divisible by 3 .


Hence 489365 is not divisible by 3 .


III) Divisibility By 9 : A number is divisible by 9 only when the Sum of its digits is


divisible by 9 .


a) In the number 246591 ,the Sum of its digits = 27 , which is divisible by 9 .


Hence 246591 is divisible by 9 .


b) In the number 437915 ,the Sum of its digits = 29 ,which is not divisible by 9.


Hence 437915 is not divisible by 9.


IV) Divisibility By 4 : A number is divisible by 4 if its last two digits is divisible by 4.


a) 8769376 is divisible by 4 , since 76 is divisible by 4.


b) 694139 is not divisible by 4 , since 39 is not divisible by 4 .


V) Divisibility By 8 : A number is divisible by 8 if the number formed by hundred's


,ten's and unit's digit of the given number is divisible by 8 .


a) In the number 76189352 , the number formed by last 3 digits , namely 352 is


divisible by 8.


Hence 76189352 is divisible by 8 .


b) In the number 756484 , the number formed by last 3 digits , namely 484 is


not divisible by 8 . Hence 756484 is not divisible by 8 .


VI ) Divisibility By 10 : A number is divisible by 10 only when its unit digit is 0.


a) 7849320 is divisible by 10 only , since its unit digit is 0 or 5 .


b) 8765138 is not divisible by 10 since its unit digit is not 0 .


VII) Divisibility By 5 : A number is divisible by 5 only when its unit digit is 0 or 5 .


a) Each of the numbers 869765 and 452365 is divisible by 5 .


VIII) A number is divisible by 11 if the difference between the Sum of its digits at


odd places and the Sum of its digits at even places is either 0 or a number


divisible by 11 .


a) Consider the number 29435417 ( Sum of its digits at odd places ) -


( Sum of its digits at even places )



= ( 7+4+3+9) - ( 1+5+4+2) = (23 - 12) = 11 ,which is divisible by 11 .


Hence 29435417 is divisible by 11*

Assorted Multiplication Rules :-

Multiply by 5: Multiply by 10 and divide by 2.


Multiply by 6: Sometimes multiplying by 3 and then 2 is easy.


Multiply by 9: Multiply by 10 and subtract the original number.


Multiply by 12: Multiply by 10 and add twice the original number.


Multiply by 13: Multiply by 3 and add 10 times original number.


Multiply by 14: Multiply by 7 and then multiply by 2


Multiply by 15: Multiply by 10 and add 5 times the original number, as above.


Multiply by 16: You can double four times, if you want to. Or you can multiply by 8 and then by 2.


Multiply by 17: Multiply by 7 and add 10 times original number.


Multiply by 18: Multiply by 20 and subtract twice the original number (which is obvious from the first step).

Multiply by 19: Multiply by 20 and subtract the original number.


Multiply by 24: Multiply by 8 and then multiply by 3.


Multiply by 27: Multiply by 30 and subtract 3 times the original number (which is obvious from the first step).

Basic Concept Of Math's .

How to multiply up to 20x20 in your head?

Assumption: You know your multiplication table reasonably well up to 10×10.

If you want to multiply - 16 x 13 .

Step 1 – Add the unit’s digit of one to the other number –

Here, add 16 + 3 = 19

Or, add 13 + 6 = 19

Step 2 – Put a zero after the number (i.e. multiply it by 10)

Here, 19 becomes 190

Step 3 – Multiply unit’s digit of both the numbers

Here, 6x3 = 18

Step 4 – Add the product to the result of Step 2

Here, 190 + 18 = 208

So Simple !

Another example, 17x19 = (17+9)*10 + (7*9) = 260 + 63 = 323

Basic Concept Of Math's

A) Natural Numbers : Counting numbers are called natural numbers .



For Ex. 1,2,3,4,5,6,7,......& so on are all natural numbers .

B) Whole Numbers : All counting numbers and 0 form the set of whole numbers .



For Ex. ,1,2,3,4,5,6,7,.....& s on are all whole numbers .



Clearly , every natural number is whole number and 0 is a whole number which is



not a natural number .



C) Integers : All counting numbers ,zero and negatives of counting numbers form



the set of integers .



For Ex. ...........-4, -3, -2,- 1, 0, 1, 2, 3, 4, 5,............. & so on are all integers .



Set of positive integers = { 1, 3, 4, 5, 6, 7,.......}



Set f negative integers = { -1,-2,-,-4,-5,-6,-7,...}



Set of all negative integers = { 0, 1, 2, 3, 4, 5, 6, 7, ........}



D) Even And Odd Numbers : -



Even Numbers : A counting number divisible by 2 is called an even number .



For Ex. 2, 4, 6, 8, 10,....& so on are all even numbers .



Odd Numbers: A counting number not divisible by 2 is called odd number .



For Ex. 1, 3, 5, 7, 9, 11,......& so on



E) Prime Numbers : A counting number is called a prime number if it has exactly



two factors , namely itself and 1 .



For Ex. 3, 5, 7, 11, 13, 1 , 19, 23, 29, 31, 37,........& so on



F) Composite Numbers : The natural numbers which are not prime , are called



composite numbers .



G) Co Primes : Two natural numbers a and b are said to be co-prime if their HCF is 1



For Ex. (2,3),(4,5),(7,9),8,11) & so on are airs of co-primes *

Basic Concept Of Math's

A) Natural Numbers : Counting numbers are called natural numbers .



For Ex. 1,2,3,4,5,6,7,......& so on are all natural numbers .


B) Whole Numbers : All counting numbers and 0 form the set of whole numbers .



For Ex. ,1,2,3,4,5,6,7,.....& s on are all whole numbers .



Clearly , every natural number is whole number and 0 is a whole number which is



not a natural number .



C) Integers : All counting numbers ,zero and negatives of counting numbers form



the set of integers .



For Ex. ...........-4, -3, -2,- 1, 0, 1, 2, 3, 4, 5,............. & so on are all integers .



Set of positive integers = { 1, 3, 4, 5, 6, 7,.......}



Set f negative integers = { -1,-2,-,-4,-5,-6,-7,...}



Set of all negative integers = { 0, 1, 2, 3, 4, 5, 6, 7, ........}



D) Even And Odd Numbers : -



Even Numbers : A counting number divisible by 2 is called an even number .



For Ex. 2, 4, 6, 8, 10,....& so on are all even numbers .



Odd Numbers: A counting number not divisible by 2 is called odd number .



For Ex. 1, 3, 5, 7, 9, 11,......& so on



E) Prime Numbers : A counting number is called a prime number if it has exactly



two factors , namely itself and 1 .



For Ex. 3, 5, 7, 11, 13, 1 , 19, 23, 29, 31, 37,........& so on



F) Composite Numbers : The natural numbers which are not prime , are called



composite numbers .



G) Co Primes : Two natural numbers a and b are said to be co-prime if their HCF is 1



For Ex. (2,3),(4,5),(7,9),8,11) & so on are airs of co-primes *

Basic Concept Of Math's

Multiplying by 9, or 99, or 999



Multiplying by 9 is really multiplying by 10-1.

So, 9x9 is just 9x(10-1) which is 9x10-9 which is 90-9 or 81.

Let's try a harder example: 46x9 = 46x10-46 = 460-46 = 414.

One more example: 68x9 = 680-68 = 612.

To multiply by 99, you multiply by 100-1.

So, 46x99 = 46x(100-1) = 4600-46 = 4554.

Multiplying by 999 is similar to multiplying by 9 and by 99.

38x999 = 38x(1000-1) = 38000-38 = 37962.

Try for another Examples.

Basic Concept Of Math's

Multiplying by 9, or 99, or 999



Multiplying by 9 is really multiplying by 10-1.

So, 9x9 is just 9x(10-1) which is 9x10-9 which is 90-9 or 81.
Let's try a harder example: 46x9 = 46x10-46 = 460-46 = 414.

One more example: 68x9 = 680-68 = 612.

To multiply by 99, you multiply by 100-1.

So, 46x99 = 46x(100-1) = 4600-46 = 4554.

Multiplying by 999 is similar to multiplying by 9 and by 99.

38x999 = 38x(1000-1) = 38000-38 = 37962.

Try for another Example s.