This chapter of Number System, simplification, and approximation is the most important chapter in whole Quantitative aptitude. This chapter forms the basis of many other topics in Quantitative **math**. You are advised to go through this chapter with the utmost care.

**Number System in Quantitative Aptitude**

Let us begin Number System by understanding various types of **numbers**.

**Natural Numbers**

The numbers which are used for counting and is denoted by N. Natural numbers can never be negative or zero.**“1 is the smallest natural number”**

Example: Natural numbers are 1,2,3,4,5,6,7,8,…..

**Is 0 a natural numbers?**

No, 0 is not considered as a natural number. In fact it is considered as whole number.

Natural numbers are further classified :

**Even Numbers:**The Numbers which are divisible by 2 are known as even numbers. For example: 2,4,6,8,10…”**2 is the smallest even number**“**Odd Numbers:**The Numbers which are not divisible by 2 are known as odd numbers. For example:1,3,5,7,9…**“1 is the smallest odd number”****Prime Numbers:**The numbers which are not divisible by any other number except 1 and itself are known as prime numbers. For example:2,3,5,7,11…**Co-prime Numbers:**Two numbers can be said to be co-prime if their highest common factor is ‘1″. For example:(4,5),(13,29) etc.**Composite Numbers:**Numbers Which are divisible by others numbers along with 1 and itself are known as composite numbers: For example: 4,6,8,10,12,14

Why 1 is not considered as prime and composite number?Prime numbers should be divisible by 1 and itself . It means we should be two divisor. In this case we have only one divisor that is 1 itself. So 1 is not considered asprime number. Composite numbers also need two divisors. In this case we have only one divisor that is 1 itself. So 1 is not considered ascomposite number.

**Whole Numbers**

All the natural numbers along with 0 are known as whole numbers. For example:0,1,2,3,4,5,6,7,8…” **0 is the smallest whole number”**

**Integers**

All counting numbers including 0 are known as integers. They are further classified into:

**Positive Integers:**All integers which are positive are known as positive integers. For example:1,2,3,4,5,6…**Negative Integers:**All integers which are negative are known as negative integers. For example:-1,-2,-3,-4,-5…

The numbers which are either rational or irrational are known as real numbers.For example: 38,0,-5,1/3,π all are real numbers. They are further classified into two types.

**Real Numbers**

**Rational Numbers:**Any numbers which can be expressed in the form of p/q where p and q both are integers and q≠0. For example:3/5,-3/5 etc.**Irrational Numbers:**The numbers which cannot be expressed in the form of p/q where p and q both are integers and q≠0 are known as irrational numbers. For example:**√**3,**√**5,π etc.

**Imaginary Numbers**

Those numbers which are non real are known as imaginary numbers. For example: **√**-5,**√**-3,**√**-7. Imaginary numbers are denoted by ‘iota'(i).

Here we will attend a short quiz of Quantitative Aptitude which will help us to understand what we have studied.

[ays_quiz id=’2′]**Division(Quantitative Aptitude)**

Let us consider A and B are two numbers then A/B is known as Division where A is the dividend and B is the divisor. Quotient, the result obtained by dividing one number by another.

**Dividend =Divisor X Quotient **

**Rules(numbers 2-8)**

**A number is divisible by 2**if the digit at unit’s place is even or zero. For example:24,486,1286 etc

**A number is divisible by 3**if the sum of all digits are divisible by 3. For example:381(3+8+1=12, 12/3=4) This number is divisible by 3.

**A number is divisible by 4**if the number formed by last two digits is divisible by 4. For example:14084(in this number 84 is completely divisible by 4 ), hence 14084 is completely divisible by 4

**A number is divisible by 5**if the digit at unit’s place is 5 or zero. For example:12500( the digit at unit’s place is 0), hence this number is completely divisible by 5.

**A number is divisible by 6**if the number is divisible by both 2 and 3. For example:1320(the number is completely divisible by 2 and 3), hence this number is completely divisible by 6.

**A number is divisible by 7**if the difference between twice the digit at unit’s place and the remaining digits is either 0 or multiple of 7.Foe example:294(4*2=8, 29-8=21 and 21 is divisible by 7), hence this number is completely divisible by 7.

**A number is divisible by 8**if the last three digits are divisible by 8. For example: 12816(816 is exactly divisible by 8) hence this number is exactly divisible by 8.

**Rules (numbers 9,10,11,12,15,19,25)**

**A number is divisible by 9**if the sum of all of its digits is completely divisible by 9. For example 1295874(the sum of its digits 1+2+9+5+8+7+4=36/9=4) hence this number is completely divisible by 9.

**A number is divisible by 10**if the digit at unit’s place is zero. For example:12580 , this number is exactly divisible by 0,because unit’s place digit is 0.

**A number is divisible by 11**if difference between the sum of digits at odd place and sum of digits at even place is 0 or 11. For example:2332(odd place=2+3=5, even place 2+3=5 so 5-5=0) hence the number is exactly divisible by 11.

**A number is divisible by 12**if the number is divisible by both 3 and 4. For example:2268(this number is divisible by both 3 and 4), hence this number is completely divisible by 12.

**A number is divisible by 15**if it is divisible by both 3 and 5. For example:1605(this number is divisible by both 3 and 5) hence this number is divisible by 15.

**A number is divisible by 19**if the sum of digits other than unit digit and twice of unit’s digit is divisible by 19. For example:76(7+2*6=19/19=1) hence this number is completely divisible by 19.

**A number is divisible by 25**if its last two digits are divisible by 25. For example: 1225(the last two digits are divisible by 25), so this number is divisible by 25.

**Quantitative Aptitude – Some key points of Number System**

- 0 is not a natural number.

- o is neither positive number nor negative number.Therefore this number is known as neutral number.

- 2 is the only even prime number.

- The sum of 1st n natural numbers=n(n+1)/2

- sum of 1st n even numbers=n(n+1)

- The sum of 1st n odd numbers=
*n*2

- The sum of square of first n natural numbers=n(n+1)(2n+1)/6

In the next post, we will discuss LCM and HCF. If you have any doubts or concern then you can mail us on [email protected]