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Operation on Numbers

Types of Numbers
Natural Numbers: The numbers 1,2,3,4.... are called natural numbers or positive integers.
Whole Numbers: The numbers 0,1,2,3.... are called whole numbers. Whole numbers include “0”.
Integers: The numbers .... -3, -2, -1, 0, 1, 2, 3,.... are called integers. You will see questions on integers in almost all the exams where you see number system aptitude questions.
Negative Integers: The numbers -1, -2, -3, ... are called negative integers.
Positive Fractions: The numbers(2/3) ,(4/5) ,(7/8) ... are called positive fractions.
Negative Fractions: The numbers -(6/8) ,-(7/19) , -(12/17) ... are called negative fractions.
Rational Numbers: Any number which is a positive or negative integer or fraction, or zero is called a rational number. A rational number is one which can be expressed in the following format ⇒(a/b) , where b ≠ 0 and a & b are positive or negative integers.
Irrational Numbers: An infinite non-recurring decimal number is known as an irrational number. These numbers cannot be expressed in the form of a proper fraction a/b where b ≠ 0. e.g.√2 , √5 , Π, etc.
Surds: Any root of a number, which cannot be exactly found is called a surd. Essentially, all surds are irrational numbers. e.g. √2 , √5 etc.
Even Numbers: The integers which are divisible by 2 are called even numbers e.g. -4, 0, 2, 16 etc.
Odd Numbers: The integers which are not divisible by 2 are odd numbers e.g. -7, -15, 5, 9 etc.
Prime Numbers: Those numbers, which are divisible only by themselves and 1, are called prime numbers. In other words, a number, which has only two factors, 1 and itself, is called a prime number. e.g. 2, 3, 5, 7, etc.
Note: 2 is the only even prime number.
There are 25 prime numbers upto 100. These are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 & 97. These should be learnt by heart.
Co-Prime Numbers: Two numbers are considered to be prime to each other if their HCF is 1. e.g. 5 and 24 are prime to each other. In other words, 5 and 24 are co-prime.
To check whether a number is prime, e.g. 79, we do not need to check all the factors below 79. The square of 8 is 64 & that of 9 is 81. Therefore, check if any of the prime numbers less than 9 is a factor of 79. The prime numbers 2, 3, 5, 7 are not the factors of 79. So, 79 is a prime number.
Composite Number: A number, which has factors other than itself and 1, is called a composite number. e.g. 9, 16, 25....
Note: 1 is neither a composite number nor a prime number.
  • Divisibility Rules:
  1. A number is divisible by 2 if it is an even number.
  2. A number is divisible by 3 if the sum of the digits is divisible by 3. 
  3. A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
  4. A number is divisible by 5 if the units digit is either 5 or 0.
  5. A number is divisible by 6 if the number is also divisible by both 2 and 3. 
  6. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
  7. A number is divisible by 9 if the sum of the digits is divisible by 9.
  8. A number is divisible by 10 if the units digit is 0. 
  9. A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places, is divisible by 11.
  10. A number is divisible by 12 if the number is also divisible by both 3 and 4. 

POINT TO REMEMBER FOR SPECIAL CASE - 

For Divisibility of 7 - We take Unit digit & multiply with 2 then Substract . 
For Divisibility of 13 - We take Unit digit & multiply with 4 then Add . 
For Divisibility of 17 - We take Unit digit & multiply with 5 then Substract . 
For Divisibility of 19 - We take Unit digit & multiply with 2 then Add . 
# A number is divisible by 7 if it Follows the below rules: 
First of all we recall the osculator for 7. Once again , for your convenience , as 7x3 = 21 (One More than 2 X 10), our negative osculator is 2 . This Osculator '2' is our key - digit . This and only this digit is used to check the divisibility of any number by 7. See how it works - Ex. Is 112 divisible by 7 ? Step I : 112 : 11 - 2x2 = 7 (Separate the last digit & multiply with two & then subtract)
Here we can see 7 Is divisible by 7, then we can say 112 is also divisible by 7.

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