# Aptitude - Number System

## Numbers

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 = 1Place value of 2 = 2 x 10

^{1}= 2 x 10 = 20Place 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...].**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...]

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

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

Every natural number is a whole number.

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.

1 is neither a prime number nor a composite number.

2 is the only even prime number.

## Divisibility

Following are tips to check divisibility of numbers.

**Divisibility by 2**- A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.**Divisibility by 3**- A number is divisible by 3 if sum of its digits is completely 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.**Divisibility by 5**- A number is divisible by 5 if its unit digit is 0 or 5.**Divisibility by 6**- A number is divisible by 6 if the number is divisible by both 2 and 3.**Divisibility by 8**- A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.**Divisibility by 9**- A number is divisible by 9 if sum of its digits is completely divisible by 9.**Divisibility by 10**- A number is divisible by 10 if its unit digit is 0.**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: 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.

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.

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.

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.

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.

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.

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.

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.

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

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.

## 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 Formulaes

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)