Aptitude - 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 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 of a digit in a numeral is value of the digit multiplied by 10n where n starts from 0. For example in 321:
Place value of 1 = 1 x 100 = 1 x 1 = 1
Place value of 1 = 1 x 100 = 1 x 1 = 1
Place value of 2 = 2 x 101 = 2 x 10 = 20
Place value of 2 = 2 x 101 = 2 x 10 = 20
Place value of 3 = 3 x 102 = 3 x 100 = 300
Place value of 3 = 3 x 102 = 3 x 100 = 300
0th position digit is called unit digit and is the most commonly used topic in aptitude tests.

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...]


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.

Natural Numbers - n > 0 where n is counting number; [1,2,3...]
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...].
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...]


0 is neither positive nor negative integer.


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...]
Positive Integers - n > 0; [1,2,3...]
Negative Integers - n < 0; [-1,-2,-3...]
Negative Integers - n < 0; [-1,-2,-3...]
Non-Positive Integers - n ≤ 0; [0,-1,-2,-3...]
Non-Positive Integers - n ≤ 0; [0,-1,-2,-3...]
Non-Negative 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,...]
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,...]
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.
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.
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.
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.
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.
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.
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.


Divisibility by 2 - A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.
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.
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.
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.
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.
Divisibility by 6 - A number is divisible by 6 if the number is divisible by both 2 and 3.
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.
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.
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.
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.
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.


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

If a number n is divisible by two co-primes numbers a, b then n is divisible by ab.
If a number n is divisible by two co-primes numbers a, b then n is divisible by ab.
(a-b) always divides (an - bn) if n is a natural number.
(a-b) always divides (an - bn) if n is a natural number.
(a+b) always divides (an - bn) if n is an even number.
(a+b) always divides (an - bn) if n is an even number.
(a+b) always divides (an + bn) if n is an odd number.
(a+b) always divides (an + bn) if n is an odd number.
When a number is divided by another number then
Following are formulaes for basic number series:

(1+2+3+...+n) = (1/2)n(n+1)
(12+22+32+...+n2) = (1/6)n(n+1)(2n+1)
(13+23+33+...+n3) = (1/4)n2(n+1)2

(1+2+3+...+n) = (1/2)n(n+1)
(1+2+3+...+n) = (1/2)n(n+1)
(12+22+32+...+n2) = (1/6)n(n+1)(2n+1)
(12+22+32+...+n2) = (1/6)n(n+1)(2n+1)
(13+23+33+...+n3) = (1/4)n2(n+1)2
(13+23+33+...+n3) = (1/4)n2(n+1)2
These are the basic formulae:
(a + b)2 = a2 + b2 + 2ab
(a - b)2 = a2 + b2 - 2ab
(a + b)2 - (a - b)2 = 4ab
(a + b)2 + (a - b)2 = 2(a2 + b2)
(a2 - b2) = (a + b)(a - b)
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
(a3 + b3) = (a + b)(a2 -  ab + b2)
(a3 - b3) = (a - b)(a2 + ab + b2)
(a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)