## Number system

In Decimal number system, there are ten symbols namely 0,1,2,3,4,5,6,7,8 and 9 called digits. A number is denoted by group of these digits called as numerals.

## Face Value

Face value of a digit in a numeral is value of the digit itself. For example in 321, face value of 1 is 1, face value of 2 is 2 and face value of 3 is 3.

## Place Value

Place value of a digit in a numeral is value of the digit multiplied by 10^{n} where n starts from 0. For example in 321:

- Place value of 1 = 1 x 10
^{0}= 1 x 1 = 1 - Place value of 2 = 2 x 10
^{1}= 2 x 10 = 20 - Place value of 3 = 3 x 10
^{2}= 3 x 100 = 300

0^{th} position digit is called unit digit and is the most commonly used topic in aptitude tests.

## Types of Numbers

**Natural Numbers**– n > 0 where n is counting number; [1,2,3…]**Whole Numbers**– n ≥ 0 where n is counting number; [0,1,2,3…].

0 is the only whole number which is not a natural number.

Every natural number is a whole number.

**Integers**– n ≥ 0 or n ≤ 0 where n is counting number;…,-3,-2,-1,0,1,2,3… are integers.

**Positive Integers**– n > 0; [1,2,3…]**Negative Integers**– n < 0; [-1,-2,-3…]**Non-Positive Integers**– n ≤ 0; [0,-1,-2,-3…]**Non-Negative Integers**– n ≥ 0; [0,1,2,3…] number system

0 is neither positive nor negative integer.

**Even Numbers**– n / 2 = 0 where n is counting number; [0,2,4,…]**Odd Numbers**– n / 2 ≠ 0 where n is counting number; [1,3,5,…]**Prime Numbers**– Numbers which is divisible by themselves only apart from 1.

1 is not a prime number.

To test a number p to be prime, find a whole number k such that k > √p. Get all prime numbers less than or equal to k and divide p with each of these prime numbers. If no number divides p exactly then p is a prime number otherwise it is not a prime number.

Example: 191 is prime number or not?

Solution:

Step 1 – 14 > √191

Step 2 – Prime numbers less than 14 are 2,3,5,7,11 and 13.

Step 3 – 191 is not divisible by any above prime number.

Result – 191 is a prime number.

Example: 187 is prime number or not?

Solution:

Step 1 – 14 > √187

Step 2 – Prime numbers less than 14 are 2,3,5,7,11 and 13.

Step 3 – 187 is divisible by 11.

Result – 187 is not a prime number.

**Composite Numbers**– Non-prime numbers > 1. For example, 4,6,8,9 etc.

1 is neither a prime number nor a composite number.

2 is the only even prime number.

**Co-Primes Numbers**– Two natural numbers are co-primes if their H.C.F. is 1. For example, (2,3), (4,5) are co-primes.

## Divisibility number system

Following are tips to check divisibility of numbers. number system

**Divisibility by 2**– A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.

Example: 64578 is divisible by 2 or not?

Solution:

Step 1 – Unit digit is 8.

Result – 64578 is divisible by 2.

Example: 64575 is divisible by 2 or not?

Solution:

Step 1 – Unit digit is 5.

Result – 64575 is not divisible by 2.

**Divisibility by 3**– A number is divisible by 3 if sum of its digits is completely divisible by 3.

Example: 64578 is divisible by 3 or not?

Solution:

Step 1 – Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30

which is divisible by 3.

Result – 64578 is divisible by 3.

Example: 64576 is divisible by 3 or not?

Solution:

Step 1 – Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28

which is not divisible by 3.

Result – 64576 is not divisible by 3.

**Divisibility by 4**– A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.

Example: 64578 is divisible by 4 or not?

Solution:

Step 1 – number formed using its last two digits is 78

which is not divisible by 4.

Result – 64578 is not divisible by 4.

Example: 64580 is divisible by 4 or not?

Solution:

Step 1 – number formed using its last two digits is 80

which is divisible by 4.

Result – 64580 is divisible by 4.

**Divisibility by 5**– A number is divisible by 5 if its unit digit is 0 or 5.

Example: 64578 is divisible by 5 or not?

Solution:

Step 1 – Unit digit is 8.

Result – 64578 is not divisible by 5.

Example: 64575 is divisible by 5 or not?

Solution:

Step 1 – Unit digit is 5.

Result – 64575 is divisible by 5.

**Divisibility by 6**– A number is divisible by 6 if the number is divisible by both 2 and 3. number system

Example: 64578 is divisible by 6 or not?

Solution:

Step 1 – Unit digit is 8. Number is divisible by 2.

Step 2 – Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30

which is divisible by 3.

Result – 64578 is divisible by 6.

Example: 64576 is divisible by 6 or not?

Solution:

Step 1 – Unit digit is 8. Number is divisible by 2.

Step 2 – Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28

which is not divisible by 3.

Result – 64576 is not divisible by 6.

**Divisibility by 8**– A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.

Example: 64578 is divisible by 8 or not?

Solution:

Step 1 – number formed using its last three digits is 578

which is not divisible by 8.

Result – 64578 is not divisible by 8.

Example: 64576 is divisible by 8 or not?

Solution:

Step 1 – number formed using its last three digits is 576

which is divisible by 8.

Result – 64576 is divisible by 8.

**Divisibility by 9**– A number is divisible by 9 if sum of its digits is completely divisible by 9.

Example: 64579 is divisible by 9 or not?

Solution:

Step 1 – Sum of its digits is 6 + 4 + 5 + 7 + 9 = 31

which is not divisible by 9.

Result – 64579 is not divisible by 9.

Example: 64575 is divisible by 9 or not?

Solution:

Step 1 – Sum of its digits is 6 + 4 + 5 + 7 + 5 = 27

which is divisible by 9.

Result – 64575 is divisible by 9.

**Divisibility by 10**– A number is divisible by 10 if its unit digit is 0.

Example: 64575 is divisible by 10 or not?

Solution:

Step 1 – Unit digit is 5.

Result – 64578 is not divisible by 10.

Example: 64570 is divisible by 10 or not?

Solution:

Step 1 – Unit digit is 0.

Result – 64570 is divisible by 10.

**Divisibility by 11**– A number is divisible by 11 if difference between sum of digits at odd places and sum of digits at even places is either 0 or is divisible by 11.

Example: 64575 is divisible by 11 or not?

Solution:

Step 1 – difference between sum of digits at odd places

and sum of digits at even places = (6+5+5) – (4+7) = 5

which is not divisible by 11.

Result – 64575 is not divisible by 11.

Example: 64075 is divisible by 11 or not?

Solution:

Step 1 – difference between sum of digits at odd places

and sum of digits at even places = (6+0+5) – (4+7) = 0.

Result – 64075 is divisible by 11.

## Tips on Division number system

- If a number n is divisible by two co-primes numbers a, b then n is divisible by ab.
- (a-b) always divides (a
^{n}– b^{n}) if n is a natural number. - (a+b) always divides (a
^{n}– b^{n}) if n is an even number. - (a+b) always divides (a
^{n}+ b^{n}) if n is an odd number. number system

## Division Algorithm

When a number is divided by another number then

**Dividend = (Divisor x Quotient) + Reminder**

## Series

Following are formulaes for basic number series:

- (1+2+3+…+n) = (1/2)n(n+1)
- (1
^{2}+2^{2}+3^{2}+…+n^{2}) = (1/6)n(n+1)(2n+1) - (1
^{3}+2^{3}+3^{3}+…+n^{3}) = (1/4)n^{2}(n+1)^{2}

## Basic Formulas

These are the basic formulae:

**(a + b) ^{2} = a^{2} + b^{2} + 2ab**

(a – b)^{2} = a^{2} + b^{2} – 2ab

(a + b)^{2} – (a – b)^{2} = 4ab

(a + b)^{2} + (a – b)^{2} = 2(a^{2} + b^{2})

(a^{2} – b^{2}) = (a + b)(a – b)

(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)

(a^{3} + b^{3}) = (a + b)(a^{2} – ab + b^{2})

(a^{3} – b^{3}) = (a – b)(a^{2} + ab + b^{2})

(a^{3} + b^{3} + c^{3} – 3abc) = (a + b + c)(a^{2} + b^{2} + c^{2} – ab – bc – ca)

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