Number System

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 10n where n starts from 0. For example in 321: 
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 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. 


Types of Numbers: 
1. Natural Numbers: A number n > 0 where n is counting number; [1,2,3...] 
2. Whole Numbers: A number 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. 
3. Integers: A number n ? 0 or n ? 0 where n is counting number;...,-3,-2,-1,0,1,2,3... are integers. 
4. Positive Integers: A number n > 0; [1,2,3...] 
5. Negative Integers: A number n < 0; [-1,-2,-3...] 
6. Non-Positive Integers: n ? 0; *0,-1,-2,-3...] 
7. Non-Negative Integers: A number n ? 0; *0,1,2,3...+ 
0 is neither positive nor negative integer. 
8. Even Numbers: A number divisible by 2; [for example 0,2,4,...] 
9. Odd Numbers: A number not divisible by 2; [for example 1,3,5,...] 
10. Prime Numbers: A number numbers which is divisible by themselves only apart from 1. 
1 is not a prime number. 


Testing of prime numbers: 
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? 
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? 
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. 

7. Composite Numbers: A number 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. 
 

8. 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. 
9. Twin prime numbers:- 
Two prime numbers A, B (A
 

Following are formulaes for basic number series: 

1.  (1+2+3+...+n)  = n/2 (n + 1) 

2.  (12+22+32+...+n2) =  n/6 * (n+1)(2n+1) 

3.  (13+23+33+...+n3) = [n(n+1)/2]2


Basic Formulaes: 
1. (a + b)2   = a2 + b2 + 2ab 
2. (a-b)2   = a2 + b2 - 2ab 
3. (a + b)2 - (a-b)2  = 4ab 
4. (a + b)2 + (a-b)2  = 2(a2 + b2)
5. (a2-b2)   = (a + b)(a-b) 
6. (a + b + c)2  = a2 + b2 + c2 + 2(ab + bc + ca) 
7. (a3 + b3. = (a + b)(a2 - ab + b2) 
8. (a3-b3)   = (a-b)(a2 + ab + b2) 
9. (a3+b3+c3-3abc)  = (a + b + c)(a2 + b2 + c2 – ab – bc-ca)  


Unit digit of sum/difference/products of numbers: 
To get last digit of numbers in sum/difference/product form just multiply the last digits of each numbers. For example last digit of 123+345+5678 is same as last digit of 3+5+8=16 i.e. 6 
And last digit of 123x34567x8739 is same as last digit of 3x7x9=189 i.e. 9  


Last digit of products of numbers having powers: 


Finding the Unit Digit of Powers of numbers having last digit as 2: 1. First of all, Divide the power of last digit of given number i.e.2 by 4. 
2. If you get any remainder, put it as the power of 2 and get the result using the below given table. 
3. If you don't get any remainder after dividing the power of 2 by 4, your answer will be (2)4 which always give 6 as the remainder 

Power -  Unit digit
21 = 2
22 = 4
23 = 8
24 = 6


(1) Find the Units Digit in (5122)24433: 

Solution: - 

Step-1: Divide the power of last digit of given number i.e.2 by 4. It means, divide 33 by 4. 
Step-2: You get remainder 1. 
Step-3: Since you have got 1 as a remainder , put it as a power of 2 i.e. (2)1. 
Step-4: Have a look on table, (2)1=2. So, Answer will be 2 


(2) Find the Unit Digit in (13452)1240: 
Solution: - 
Step-1: Divide the power of last digit of given number i.e.2 by 4. It means, divide 40 by 4. 
Step-2: It's completely divisible by 4. It means, the remainder is 0. 
Step-3: Since you have got nothing as a remainder , put 4 as a power of 2 i.e. (2)4. 
Step-4: Have a look on table, (2)4=6. So, Answer will be 6 


Finding the Unit Digit of Powers of numbers having last digit as 3: 
1. First of all, Divide the power of last digit of given number i.e.3 by 4. 
2. If you get any remainder, put it as the power of 3 and get the result using the below given table. 
3. If you don't get any remainder after dividing the power of 3 by 4, your answer will be (3)4 which always give 1 as the remainder 

31 = 3
32 = 9
33 = 7
34 = 1


(1) Find the Units Digit in (123)346433: 
Solution: - 
Step-1: Divide the power of last digit of given number i.e.3 by 4. It means, divide 33 by 4. 
Step-2: You get remainder 1. 
Step-3: Since you have got 1 as a remainder , put it as a power of 3 i.e. (3)1. 
Step-4: Have a look on table, (3)1=3. So, Answer will be 3 


(2) Find the Unit Digit in (1453)25632: 
Solution: - 
Step-1: Divide the power of last digit of given number i.e.3 by 4. It means, divide 32 by 4. 
Step-2: It's completely divisible by 4. It means, the remainder is 0. 
Step-3: Since you have got nothing as a remainder , put 4 as a power of 3 i.e. (3)4. 
Step-4: Have a look on table, (3)4=1. So, Answer will be 1 


Finding the Unit Digit of Powers of numbers having last digit 0,1,5,6: The unit digit of 0,1,5,6 always remains same i.e. 0,1,5,6 respectively for every power. 
Finding the Unit Digit of Powers of 4 & 9 
In case of 4 & 9, if powers are Even, the result will be 6 & 4. However, when their powers are Odd, the result will be 1 & 9. The same is depicted below: 
- If the Power of 4 is Even, the result will be 6 
- If the Power of 4 is Odd, the result will be 4 
- If the Power of 9 is Even, the result will be 1 
- If the Power of 9 is Odd, the result will be 9. 

For Example -  

 
 
 
- (119)1684 = 1 
- (239)2421 = 9 
- (564)3264 = 6 
- (874)4463 = 4 


Finding the Unit Digit of Powers of numbers having last digit as 7: 1. First of all, Divide the power of last digit of given number i.e.7 by 4. 

2. If you get any remainder, put it as the power of 7 and get the result using the below given table. 
3. If you don't get any remainder after dividing the power of 7 by 4, your answer will be (7)4 which always give 1 as the remainder 

71 = 7
72 = 9
73 = 3
74 = 1


(1) Find the Units Digit in (987)5234 

Solution: - 
Step-1: Divide the power of last digit of given number i.e.7 by 4. It means, divide 5234 by 4. 
Step-2: You get remainder 2. 
Step-3: Since you have got 2 as a remainder , put it as a power of 7 i.e. (7)2. 
Step-4: Have a look on table, (7)2=9. So, Answer will be 9 



(2) Find the Unit Digit in (5647)81284 

Solution: - 
Step-1:: Divide the power of last digit of given number i.e.7 by 4. It means, divide 84 by 4. 
Step-2: It's completely divisible by 4. It means, the remainder is 0. 
Step-3: Since you have got nothing as a remainder , put 4 as a power of 7 i.e. (7)4. 
Step-4: Have a look on table, (7)4=1. So, Answer will be 1 
 

Finding the Unit Digit of Powers of numbers having last digit as 8: 1. First of all, Divide the power of last digit of given number i.e.8 by 4. 
2. If you get any remainder, put it as the power of 8 and get the result using the below given table. 
3. If you don't get any remainder after dividing the power of 8 by 4, your answer will be (8)4 which always give 6 as the remainder 

81 = 8
82= 4
83 = 2
84 = 6


(1) Find the Units Digit in (1238)31234: 

Solution: - 

Step-1:: Divide the power of last digit of given number i.e.1238 by 4. It means, divide 31234 by 4. 
Step-2: You get remainder 2. 
Step-3: Since you have got 2 as a remainder , put it as a power of 8 i.e. (8)2. 
Step-4: Have a look on table, (8)2=4. So, Answer will be 4 

(2) Find the Unit Digit in (78658)36032: 

Solution: - 
Step-1:: Divide the power of last digit of given number i.e.8 by 4. It means, divide 36032 by 4. 
Step-2: It's completely divisible by 4. It means, the remainder is 0. 
Step-3: Since you have got nothing as a remainder , put 4 as a power of 8 i.e. (8)4. 
Step-4: Have a look on table, (8)4=1. So, Answer will be 6.