Quantitative Aptitude - Number System

Properties of real numbers

In Hindu Arabic system we use ten symbols 0,1,2,3,4,5,6,7,8,9 called digits to represent any number.
This is the decimal system where we use the digits 0 to 9.
Here o is called insignificant digit where as 1,2,3,4,5,6,7,8,9 are significant digits

Number systems
Natural numbers : The numbers 1,2,3,4,5,.......125,126, ........... which we use in counting are known as natural numbers
The set of all natural numbers can be represented by N = {1,2,3,4,5 ........... }

Whole Numbers :-
If we include 0 among the natural numbers then the numbers 0,1,2,3,4,5 ........... are called whole numbers
The set of whole numbers can be represented by W = {0,1,2,3,4,5, ........... .}
Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number

Integers All counting numbers and their negatives including zero are known as integers.
The set of integers can be represented by z or I = {.........-4,-3,-2,-1,0,1,2,3,4..}
Clearly every natural number is an integer but not every integer is natural number

Positive Integers
The set I + = {1,2,3,4........} is the set of all positive integers.
Clearly positive integers and natural numbers are synonyms

Negative Integers
The set I - = {-1,-2,-3,.......} is the set of all negative integers
0 is neither positive nor negative.

Non negative integers :
The set {0,1,2,3........} is the set of all non negative integers

Rational Numbers
The numbers of the form P/q, where p and q are integers and q≠0, are known as rational numbers. e.g:3/7, 5/2, -5/9, 1/2, -3/5 etc ,e.g  9/0 is not define
The set of all rational numbers is denoted by Q.

Example: Q = { x :x = p/q; where q ≠ 0}

Irrational Numbers
Those numbers which when expressed in decimal form are neither terminating nor repeating decimals are known as irrational numbers e.g = The absolute value of 10/3, 22/7 & 2^1/2 etc
The absolute value of P is irrational. 22/7 is rational.

Real Numbers
The rational and irrational numbers together are called real numbers.

Example: 13/21, 2/5, -3/7, / 3, +4 / 2 etc are real numbers
The set of real numbers is denoted by R
 
Even Numbers
All those natural numbers which are exactly divisible by 2 are called even numbers

Example: 2,6,8,10 etc are even numbers.
 
Odd Numbers
All those numbers which are not exactly divisible by 2 are called odd numbers.

Example: 1,3,5,7. Etc are odd numbers
 
Prime Numbers
The natural numbers other than 1, is a prime number if it is divisible by 1 and it self only

Example: Each of the numbers 2,3,5,7,11 ,13,17,etc are prime number.
1 Is not a prime number
2 Is the least and only even prime number
3 Is the least odd prime number
Prime numbers upto 100 are
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,57,61,67,71,73,79,83,89,97.

Example 25 prime numbers

Composite Number
Natural numbers greater than 1 which are not prime, are known as composite numbers
The number 1 is neither prime number nor composite number
Two numbers which have only '1' the common factor are called co-primes (or) relatively prime to each other

Example : 3 and 5 are co primes

Composite Number
Natural numbers greater than 1 which are not prime, are known as composite numbers
The number 1 is neither prime number nor composite number
Two numbers which have only '1' the common factor are called co-primes (or) relatively prime to each other

Example : 3 and 5 are co primes

Face Value and Place Value of a Digit

Face Value: It is the value of the digit itself eg, in 3452, face value of 4 is ‘four’, face value of 2 is ‘two’.

Place Value: It is the face value of the digit multiplied by the place value at which it is situated eg, in 2586, place value of 5 is 5 × 102 = 500

Tests of Divisibility

Divisibility by 2:
A number is divisible by 2 if the unit's digit is either zero or divisible by2.

Example: units digit of 76 is 6 which is divisible by 2 hence 76 is divisible by 2 and units digit of 330 is 0 and hence it is divisible by 2
 
Divisibility by 3
A number is divisible by 3 if the sum of all digits in it is divisible by 3.

Example:  The number 273 is divisible by 3 since 2+ 7 + 3 = 12 which is divisible by 3.

Divisibility by 4
A number is divisible by 4, if the number formed by the last two digits in it is divisible by 4, or both the last digits are zeros

Example: The number 5004 is divisible by 4 since last two digits 04 is divisible by 4.
 
Divisibility by 5
A number is divisible by 5 if the units digit in the number is either 0 or 5

Example: 375 is divisible by 5 as 5 is present in units place in the number.
 
Divisibility by 6
A number is divisible by 6 if the number is even and sum of all its digits is divisible by 3

Example: The number 6492 is divisible as it is even and sum of its digits 6 + 4 + 9 + 2 = 21 is divisible by 3
 
Divisibility by 7
A number is divisible by 7 if the difference of number' obtained by
omitting the unit digit' and 'twice the units digits' of the given number is divisible by 7

Example: Consider the number 10717
On doubling the unit digit '7' we get 14.
On omitting the unit digit of 10717 we get 1057.
Then 1071 -14= 1057 is divisible by 7
Therefore 10717 is divisible by 7
 
Divisibility by 8
A number is divisible by 8,if the number formed by last 3 digits is divisible by 8.

Example: The number 6573392 is divisible by 8 as the last 3 digits '392' is divisible by 8.
 
Divisibility by 9
A number is divisible by 9 if the sum of its digit is divisible by 9

Example: The number 15606 is divisible by 9 as the sum of the digits
1 + 5 + 6 + 0 + 6 = 18 is divisible by 9.
 
Divisibility by 10
A number is divisible by 10, if it ends in zero

Example:  The last digit of 4470 is zero, therefore 4470 is divisible by 10.
 
Divisibility by 11
A number is divisible by 11 if the difference of the Sum of the digits at odd places and sum of the digits at the even places is either zero or divisible by 11.

Example: In the number 9823, the sum of the digits at odd places is 9 + 2 = 11 and the sum of the digits at even places is 8 + 3= 11.
The difference between it is 11 - 11 = 0
the given number is divisible by 11.
 
Divisibility by 12
A number is divisible by 12 if it is divisible by 3 and 4

Example: The number 1644 is divisible by 12 as it is divisible by 3 and 4
 
Divisibility by 18
An even number satisfying the divisibility test by 9 is divisible by 18.

Example: The number 80388 is divisible by 18 as it satisfies the divisibility test of 9.
 
Divisibility by 25
A number is divisible by 25 if the number formed by the last two digits is divisible by 25 or the last two digits are zero

Example: The number 7975 is divisible by 25 as the last two digits are divisible by 25.
 
Divisibility by 88
A number is divisible by 88 if it divisible by 11 and 8

Example: The number 10824 is divisible by 88 as it is divisible by both 11 and 8
 
Divisibility by 125
A number is divisible by 125 if the number formed by last three digits is divisible by 125 or the last three digits are zero

Example: ' 43453375' is divisible by 125 as the last three digits '375' are divisible by 125.

‘Smart’ Facts

• If p and q are co-primes and both are factors of a number K, then their product p x q will also be a factor of r. eg, Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 prime factors of 24 are 2 and 3, which are co-prime also. Product of 2 × 3 = 6, 6 is also a factor of 24.
• If ‘p’ divides ‘q’ and ‘r’, then p’ also divides their sum or difference.

Example:  4 divides 12 and 20. Sum of 12 and 20 is 32 which is divisible by 4. Difference of 20 and 12 is 8 which is divisible by 4.

• If a number is divisible by another number, then it must be divisible by each of the factors of that number. 48 is divisible by 12. Factors of 12 are 1, 2, 3, 4, 6, 12. So, 48 is divisible by 2, 3, 4 and 6 also.

Division on Numbers

In a sum of division, we have four quantities.
They are (i) Dividend, (ii) Divisor, (iii) Quotient and (iv) Remainder. These quantities are connected by a relation.

(a) Dividend = Divisor × Quotient + Remainder.
(b) Divisor = (Dividend – Remainder) ÷ Quotient.
(c) Quotient = (Dividend – Remainder) – Divisor.

SIMPLIFICATION

In simplification we are supposed to follow the order which is essentially demanded by mathematics and given by a common note of remembrance as VBODMAS

Where
V indicates Vinculum
B indicates 'Bracket'
O indicates 'Of'
D indicates 'Division
M indicates 'Multiplication'
A indicates 'Addition'
S indicates 'Subtraction'

·        Sum of natural numbers from 1 to n = n(n+ 1)/2

Example: Sum of natural numbers from 1 to 40 = 40(40 +1)/2 = 80

• Sum of even numbers from 1 to n is k ( k +1) : where k indicates number of even numbers from 1 to n
  Example: Sum of even number from 1 to 80 is 40 (40+1) = 1640 Here from 1 to 80 there exists 40 even numbers

• Sum of odd numbers from 1 to n is ( Number of odd numbers from 1 to n ) ²
  e.g Sum of odd numbers from 1 to 60 is ( 30) ² = 900 Here from 1 to 60 there exists 30 odd natural numbers

• Sum of squares of first n natural numbers is[n(n+1) (2n+1)/6]

 Example: Sum of squares of first 20 natural numbers is [20(20+1) (40+1)/6=2870

• Sum of the squares of first n even natural numbers is [2/3n (n+1) (2n+1)
  e.g Sum of squares of first 10 even natural numbers is (2/3) x 10 x 11 x 21 = 1540
• Sum of squares first n odd natural numbers is n(2n+1) (2n-1)/3

Example: Sum of squares first 30 odd numbers is [(30)(61)(59)]/3 = 35990

• Sum of cubes of first n natural numbers is [n(n+1)/2]^2 ].

Example: Sum cubes of first 9 natural numbers is [9(9+1)/2]^2 =4050

Use of Algebraic identities : A student can use algebraic identities given below in the simplification
 
The Algebraic identities are
1.(a + b)² = a² + 2ab + b²
2.(a - b)² = a² - 2ab + b²
3.(a + b)² + (a - b)² = 2(a² + b² )
4.(a + b)² - (a - b)² = 4ab
5.a² - b² = (a + b) (a - b)
6.(a + b)³ = a³ + 3a²b +3ab² + b³ = a³ + b³ + 3ab(a + b)
7.(a - b)³ = a³ - 3a² b + 3ab² - b³ = a³ - b³ - 3ab(a - b)
8. a³ + b³ = (a + b) (a² - ab + b² )
9. a³ - b³ = (a - b) (a² + ab + b² )
10 .a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca )

Factors and Multiples

·         Factor: A number which divides a given number exactly is called a factor of the given number,
Example: 24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6

·         Thus, 1, 2, 3, 4, 6, 8, 12 and 24 are factors of 24.
• 1 is a factor of every number
• A number is a factor of itself
• The smallest factor of a given number is 1 and the greatest factor is the number itself.
• If a number is divided by any of its factors, the remainder is always zero.
• Every factor of a number is either less than or at the most equal to the given number.
• Number of factors of a number are finite.

·         Number of Factors of a Number: If N is a composite number such that N = a^m bⁿ c^o... where a, b, c ... are prime factors of N and m, n, o ... are positive integers, then the number of factors of N is given by the expression (m + 1) (n + 1) (o + 1)

Example : Find the number of factors that 224 has.
Solution. 224 = 25 × 71
Hence, 224 has (5 + 1) (1 + 1) = 6 × 2 = 12 factors.

Types of Questions asked in SSC CGL in Previous Years - 

Type I

1. Which of the following fractions is the smallest?

9/13, 17/26,28/29,33/52

(1) 33/52 (2) 17/26

(3) 9/13 (4) 28/29

Answers:

(1) 9/13 = 0.69

17/26 =0.65

28/29= 0.96

33/52 = 0.63

2. The least among the fractions 15/16,19/20,24/25,34/35 is

(a) 34/35 (2) 15/16

(3) 19/20 (4) 24/25

Answers: 

(2) : 

15/16 = 0.94

19/20=0.95

24/25-0.96

34/35=0.97

Type - II

3. A number when divided by 899 gives a remainder 63. If the same number is divided by 29, the remainder will be: 

(a) 10 (2) 5

(c) 4 (4) 2

Answers: (2)

Required remainder = remainder got when 63 is divided by 29 = 5

4. In a question on division, the divisor is 7 times the quotient and 3 times the remainder. If the remainder is 28, then the dividend is 

(1) 588 (2) 784

(3) 823 (4) 1036

Answers:

(4) Let the quotient be Q and the remainder be R. Then 

Divisor = 7Q = 3R

Q = 3/7 R = (3/7) x 28 = 12

Divisor = 7Q = 7 x 12 = 84 

Dividend = Divisor x Quotient + Reminder = 84 x 12 + 28 = 1008 + 28 = 1036

Type - III

5. A person goes 1/4  of the property to his daughter, 1/2 to his sons and 1/5 for charity. How much has he given away?

(1) 1/20 (2) 19/20

(3) 1/10 (4) 9/10

Answers: (2)

Part of the property given away = 1/4 + 1/2 + 1/5 = 19/20

6. A boy was asked to find 3/5 of a fraction. Instead, he divided the fraction by 3/5 and got an answer which exceeded the correct answer by 32/75. The correct answer is 

(1) 3/25 (2) 6/25

(3) 2/25 (4) 2/15

Answers: Let the fraction be x.

According to the question

= 5x/3 – 3x/5 = 32/75

16x/15 = 32/75

X = 2/5

Correct answer = (2/5) x (3/5) = 6/25

Type - IV

7. The number 0.121212………. in the form p/q is equal to

(1) 4/11 (2) 2/11

(3) 4/33 (4) 2/33

Answers:(3):  0.121212 = 12/99 = 4/33

8. The value of 

1/15 + 1/35+ 1/63 + 1/99 + 1/143 is 

(1) 5/39 (2) 4/39

(3) 2/39 (4) 7/39

Answers: (1)  

Type - V

9. Arrange 4/5, 7/8 , 6/7, 5/6 in the ascending order.

(1) 4/5, 7/8 , 6/7, 5/6

(2) 5/6,6/7,7/8,4/5

(3) 4/5,5/6,6/7,7/8

(4) 7/8,6/7,5/6,4/5

Answers: (3) Firstly, we express every fraction in decimal form.

4/5 = 0.8

7/8 = 0.875

6/7 = 0.857

5/6 = 0.833 = .83

10. The digit in unit’s place of the product (2153)167 is :

(1) 1 (2) 3

(3) 7 (4) 9

Answers: (3) Unit’s digit in 34 = 1

So units digit in 3164 = 1

Now, unit’s digit in (2153)167

= Unit’s digit in 3167

= Unit’s digit in 33 =7

Type - VI

11. The digit in the unit’s place of [(251)]98 + (21)29 – (106)100 + ( 705)35 – 164 + 259] is:

(1) 1 (2) 4

(3) 5 (4) 6

Answers: Required answer = 1 + 1 -6 + 5 – 6 +9 = 16-12=4

12. The unit digit in the sum of (124)372 + (124)373 is 

(1) 5 (2) 4

(3) 2 (4) 0 

Answers: 41 =4; 42 =16; 43 = 64; 44= 256; 45 = 1024

Remainder on dividing 372 by 4 = 0

Remainder on dividing 373 by 4= 1

Required unit digit = Unit’s digit of the sum of 6 + 4 = 0

Type - VII

13. The sum of the square of three consecutive natural numbers is 2030. Then, what is the middle number?

(1) 25 (2) 26

(3) 27 (4) 28

Answers: (2) Let the three consecutive natural numbers be x.

x+1 and x+2

According to question.

X2 (x+1)2 + (x+2)2 = 2030 

x = 25 and x = -27

Required number = x+1

= 25 +1 = 26

14. Out of six consecutive natural numbers. If the sum of first three is 27. What is the sum of the other three?

(1) 36 (2) 35

(3) 25 (4) 24

Answers: (1) 8+9+10= 27

11+12+13 = 36