Number System Important Points

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A collection of numbers that forms a system is called Number System.

Questions asked in SSC exams related to Number System are based on :

TYPE-1: Question-based on smallest and largest fraction.
TYPE-2 : Question based on Division, Multiplication,
              Addition and Subtraction.
TYPE-3: Question-based on the sum of consecutive numbers.
TYPE-4: Question-based on fractions.
TYPE-5: Question-based on finding the ascending and descending orders.
TYPE-6: Question-based on finding the unit’s digits of a number.

Number System

Numbers are divided into following types :

Natural Numbers    –  All Counting numbers are called natural numbers.
e.g; 1,2,3,4,….etc

Whole Numbers – All counting numbers together with ZERO are called whole Numbers.
e.g; 0,1,2,3,4,…etc

NOTE: (i) 0 is a whole number which is not a natural number.
            (ii) Every natural number is a whole number.

Integers – All counting numbers, 0 and negative of counting numbers are called integers.

(a) Positiev integers : {1,2,3,4,…..}
(b) Non-Positive integers : {0,-1,-2,-3,-4,…}
(c) Negative integers : {-1,-2,-3,-4,…}
(d) Non-Negative integers : {0,1,2,3,4,….}

Even Numbers – A number which is divisible by 2 are called even number.
e.g 0,2,4,6,8,….etc.

Odd Numbers – A number which is not divisible by 3 are called odd number.
e.g 1,3,5,7,…. etc.

Prime numbers – A number which is greater than 1 and having exactly two factors namely
1 and itself are called prime number.
e.g 2,3,5,7,….etc.

Composite number – Number which is greater than 1 and is not prime is called
Composite numbers.
e.g 6,8,10,12,14,…etc.

NOTE: (a) 2 is only an even number which is prime
            (b) 1 is neither prime nor composite.

Co-Prime Numbers – Two natural numbers “A” and “B” are co-prime if: their H.C.F is 1
e.g (2,3),(4,5),(7,9),(8,11),…etc.

Consecutive Numbers – A series of numbers in which each is greater than that
which precedes it by 1 is called consecutive number.
e.g 6,7,8 or 13,14,15,16 etc

Rational Number – The number which can be written in the form of p/q
where p and q are integers and q ≠ 0
are called rational number.
e.g 1/2,3/4,5/10,… etc.

Irrational Numbers – The number which cannot be written in the form p/q
where p and q are integers and q ≠ 0
are known as an irrational number
e.g √3, √2

Real Number – The number which includes rational as well as irrational are called real number.

Key points to check divisibility :

Divisible by 2: Any number is divisible by 2 if the last digit is either Zero or an even number.
e.g 10,12,14,….etc

Here if we take 10 and 12 the last digit of 10 is “0” and 12 last digit is 2 which is even.
It means by the rules both are by 2.

 

Divisible by 3: If the sum of digit of a number is divisible by 3, then the number is also
divisible by 3.
e.g 33,36,39,333,393939, … etc

Here in 33, the sum of a digit is 3+3 = 6, and 6 is divisible by 3 that means the whole number is
divisible by 3.
we take the larger number like 333, the sum of 333 is 3+3+3 = 9, and 9 is divisible by 3

if we take a larger number like 393939, the sum of 393939 is 3+9+3+9+3+9 = 36

Now, we have to add the digit of 36 further like 3+6 = 9, we get 9 which is divisible by 3
that means the whole number 393939 is divisible by 3.

 

Divisibility by 4: If the last two digit of a number is divisible by 4. The number has two or more zero at
the end of the number is also divisible by 4.
e.g 124,100,.. etc.

Here in 124 the last digit that is 24 which is divisible by 4, It means the whole 124 is divisible by 4.
and same as 100 there are two zero at the end of the digit, and according to the definition, it is divisible by 4.

 

Divisible by 5: If any number ends with 5 or 0 then the number is divisible by 5.
e.g 10,15,20,100,200,… etc.

 

Divisible by 6: If the number is divisible by 2 and 3 both then the number is divisible by 6 also.
e.g 12,18,24,30,300,… etc.
Here 12 is divisible by both 2 as well as 3, Hence it is divisible by 6.

 

General Rules for solving problems in Arithmetic:

  1. (a+b)(a-b) = a² – b²   or   a² – b² / a + b = a – b
  2. (a-b)²        =   a² – 2ab +  b²
  3. (a+b)²      =  a² + 2ab +  b²
  4. (a+b)³     = a³ + b³ + 3a²b + 3ab² or a³ + b³ + 3ab(a+b)
  5. (a-b)³        = a³ – b³ – 3a²b + 3ab² or a³  – b³ – 3ab(a-b)
  6. a³ + b³    = (a + b)(a² – ab +  b²)
  7. a³ – b³     = (a – b)(a² + ab +  b²)

Some Rules of Counting Numbers :

1) Sum of all first n natural numbers = n(n + 1)/2
e.g ; 1+2+3+….+105 = 105(105 + 1)/2 = 5565

2) The Sum of first n odd numbers = n²
e.g ; 1+3+5+7 = 4² = 16 (as the fourth number is odd)

3)Sum of first n even numbers = n(n + 1)

e.g ; 2+4+6+8+…+50 = 50(50 + 1)
= 2550

4) Sum of square of first n natural numbers = n(n + 1)(2n + 1)/6
e.g ; 1²+2²+3² + …+10² = 10(10 + 1)(2×10 + 1) / 6 = 385

5) Sum of cubes of first n natural numbers = [n(n + 1) / 2]²
e.g ;  1³+2³+3³+…..+6³ = [6(6 + 1) / 2]² = (21)² = 441

Some Tips & Tricks

Rule 1 : When the numerator and demominatio of the fraction
increase by a constant value,then the last fraction
is the greater.
e.g ; which one is the bigger factor 3/4, 4/5 and 5/6
Here the last factor is 5/6, and hence it is greator.

Rule 2: The fraction whose numerator after cross-multiplication
gives the greater value.
e.g; Which is greater 5/8 or 9/4?
Do the cross-multiplication of the fractions
5×14 = 70 and 8×9 = 72
Here we see that 72 > 70 and the numerator involve in greater value is 9
Hence, 9/4 is a greater fraction.

Some important Question of Number System and their solutions.

Que 1. A man engaged a servant on the condition that he would pay him
Rs 90 and a turban after service of one year. He served only for nine
months and received the turban and an amount of Rs 65.
The Price of Turban is?

(a) Rs 20
(b) Rs 18
(c) Rs 10
(d) Rs 11

Ans is (c) Rs 10

12 month salery = Rs 90 + turban
∴ 9 months’ salety = (Rs 90 + turban) x 9/12
= Rs 90×3/4 + 3/4 turban
= Rs 135/2 + 3/4 turban
∴  Rs 135/2 + 3/4 turban
= Rs 65 + turban
∴ 1/4 turban = 135/2 – 65 = Rs 5/2
∴ Turban = 5/2 x 4 = Rs 10

 

Que 2 . Which of the following fraction is the smallest?
7/6 , 7/9 , 4/5 , 5/7

(a) 7/6
(b) 4/5
(c) 5/7
(d) 7/9

Ans is (c) 5/7

Sol. 7/9 = 0.77
4/5 = 0.8
5/7 = 0.71
7/9 = 1.16
Therefore 0.71 is smallest  ∴ 5/7 is smallest

Que 3. If a number is as much greater than 31 as it is less then
75, then the number is

(a) 106
(b) 44
(c) 74
(d) 53

Ans. is (d) 53

Sol. if the number be x, then
x – 31 = 75 – x
⇒ 2x = 75 + 31 = 106
∴ x = 53

Que 4. 999 x 998 / 999 x 999 = ?

(a) 998999
(b) 999899
(c) 989999
(d) 999989

Ans is (a)
sol. (999 x 998 / 999 ) x 999
= (999 + 998/999) x 999

= 999² + 998
= (1000 – 1)² + 998
= 1000000 – 2000 1 998
= 998999

Que 5. The number which is to be added to 0.01 to get 1.1 is

(a) 1.11
(b) 1.09
(c) 1
(d) 0.10

Ans. is (b) 1.09

Sol. Required number = 1.1 – 0.01 = 1.09

 

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Also read L.C.M & H.C.F

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