Number System in Maths Part-2

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 Number System in Maths:Part-2

number system, number system in maths, number system formula
Number System in Maths
Table of Contents:

Before reading this post (Number System in Maths Part-2), I would suggest you to read our previous article, link for which is mentioned below. 

Number System in Maths: Basic to Advance Formulae & Tricks

Number System in Maths:Part-2

As you know Number System is one of the largest chapter in maths that can't be cover in a single post, that's why we've divided it in various parts so that we can cover every sections of this chapter.

Do you know? We can boost our aptitude solving speed 10x by memorizing the value of square root up to 10, perfect square up to 40 and perfect cube up to 20.

Square Root Value up to 10

✓2=1.414

✓3=1.732

✓5=2.23

✓6=2.44

✓7=2.64

✓8=2.82

✓10=3.16

Perfect Square from 11 to 40

(11)²= 121        (21)²= 441            (31)²= 961

(12)²= 144        (22)²= 484            (32)²= 1024

(13)²= 169        (23)²= 529            (33)²= 1089

(14)²= 196        (24)²= 576            (34)²= 1156

(15)²= 225        (25)²= 625            (35)²= 1225

(16)²= 256        (26)²= 676            (36)²= 1296

(17)²= 289        (27)²= 729            (37)²= 1369

(18)²= 324        (28)²= 784            (38)²= 1444

(19)²= 361        (29)²= 841            (39)²= 1521

(20)²= 400        (30)²= 900            (40)²= 1600

Perfect cube from 2 to 20

(2)³= 8                             (11)³= 1331 

(3)³= 27                           (12)³= 1728

(4)³= 64                           (13)³= 2197 

(5)³= 125                         (14)³= 2744

(6)³= 216                         (15)³= 3375

(7)³= 343                          (16)³= 4096

(8)³= 512                          (17)³= 4913

(9)³= 729                          (18)³= 5832

(10)³= 1000                      (19)³= 6859     

Mathematical Formula

1. (a + b)² = a² + 2ab + b²
                 = (a — b)² + 4ab
2. (a — b)² = a²— 2ab + b²
                 = (a + b)² — 4ab
3. (a²— b²) = (a + b)(a — b)
4. (a²+ b²) = (a + b)² — 2ab
                  = (a — b)²+ 2ab  
5. (a + b)³ = a³+ 3a²b + 3ab²+ b³   
                 = a³+ b³ + 3ab(a + b)
6. (a — b)³ = a³— 3a²b + 3ab²— b³   
                 = a³— b³ —3ab(a— b)
7.  a³ + b³ = (a + b)(a²— ab + b²)  
                = (a + b)³— 3ab(a + b) 
8.  a³— b³ = (a — b)(a²+ ab + b²)  
                = (a — b)³+ 3ab(a — b)
9.  (a + b + c)²= a² + b² + c² + 2(ab + bc + ca)
                      = a² + b² + c² + 2abc(1/a + 1/b + 1/c)
10.  a² + b² + c²— ab —bc —ca= 1/2[(a—b)²+(b—c)²+(c—a)²]

11.  a³ + b³ + c³—3abc= (a + b + c)(a² + b² + c² — ab — bc — ca)
                                   = 1/2(a + b + c)[(a—b)²+(b—c)²+(c—a)²]
      If  (a + b + c) = 0 then   a³ + b³ + c³ = 3abc 
12.  a⁴— b⁴= (a² + b²)(a² — b²) = (a² + b²)(a + b)(a — b)
13.  If a, b ≥ 0 then (a— b) = (√a +√b)(√a —√b)   

What is Unit Digit in Number System? 

Units digit of a number is the digit in the one's place of the number. It is the last digit of a number. For an example unit digit of 567 is 7 & unit digit of 56 is 6.

Trick to find Unit Digit 0f 0, 1, 5 or 6? 

When a number having unit digit as 0, 1, 5 or 6 and is raised to some power, the unit digit obtained here will be the same number i.e. 0, 1, 5 or 6 respectively, no matter what power it is raised to.

For Example:

Q. Find the unit digit of (126)¹²³, (131)¹² & (125)¹³

Unit digit of (126)¹²³ = 6 

Unit digit of (131)¹² = 1

Unit digit of (125)¹³ = 5

Trick to find Unit Digit of 2, 3, 7 or 8? 

To find the unit digit 2, 3, 7 and 8 we divide the power by 4 and raise the unit of the original number to the remainder. But, when the power is completely divisible by 4, the unit digit is obtained by raising the unit digit of the original number to power 4.

For Example:

Q. Find the unit digit of (162)¹²³ and (648)²³¹

(162)¹²³ = Divide 123 by 4 = We get remainder 3

Now, put the remainder as the power of last digit i.e. (2)³= 8

So, the unit digit of (162)¹²³ = 8 Ans.

(648)²³¹ =  Divide 231 by 4 = We get remainder 3

Now, put the remainder as the power of last digit i.e. (8)³= 512

So, the unit digit of (648)²³¹ = 2 Ans.

Trick to find Unit Digit of 4?

If the power is odd = Unit digit will be 4
If the power is even = Unit digit will be 6

Trick to find Unit Digit of 9?

If the power is odd = Unit digit will be 9
If the power is even = Unit digit will be 1


Remainder Theorem

a) x\nolimits^n + y\nolimits^n is exactly divisible by x + y only when n is odd.
For Example:  {a^7} + {b^7}  is exactly divisible by {a^{}} + b

b) x\nolimits^n + y\nolimits^n is exactly not divisible by x + y when n is even.
For Example: {a^{10}} + {b^{10}} is exactly not divisible by {a^{}} + b.

c) x\nolimits^n + y\nolimits^n is never divisible by x - y
d) {x^n} - {y^n} is exactly divisible by x + y when n is even.
For Example: {x^{10}} - {y^{10}} is exactly divisible by x + {y^{}}

e) {x^n} - {y^n} is exactly divisible by x - y irrespective of n being odd or even.

Questions Related to Remainder Theorem

Q.1) [{49^{15}} - 1] is exactly divisible by?
Solution: ({7^{30}} - 1) is exactly divisible by ({7^{}} + 1) i.e. 8 Ans.
Q.2) It is given that ({2^{32}} + 1) is exactly divisible by a certain number. Which one of the following is also divisible by the same number?
a. ({2^{96}} + 1)  b. ({2^{16}} - 1)   c. ({2^{16}} + 1)  d. 7 \times {2^{33}}

Solution:
({2^{96}} + 1) = {({2^{32}})^3} + 1 
           = ({2^{32}} + 1) \{ {({2^{32}})^2} - {2^{32}} + 1\}
Hence, ({2^{96}} + 1) is divisible by ({2^{32}} + 1) Option. a

Q.3) Find the common factors of ({47^{43}} + {43^{43}}) & ({47^{47}} + {43^{47}})
Solution:
({47^{43}} + {43^{43}}) = 47 + 43 =90
({47^{47}} + {43^{47}}) = 47 + 43 =90

Q.4) Find the remainder of \frac{{{x^2} - 7x + 15}}{{x - 3}} ?

Solution: x - 3 = 0 \Rightarrow x = 0
Putting the value of x=0 in {x^2} - 7x + 15, we get Ans. 3
Q.5) Find the remainder of \frac{{{x^{11}} + 1}}{{x + 1}} ?
Solution: x + 1 = 0 \Rightarrow x = - 1, 
Putting the value of x in {x^{11}} + 1, we get the Ans. 0
Q.6) Find the remainder of \frac{{{x^{40}} + 3}}{{{x^4} + 1}} ?
Solution: {x^4} + 1 = 0 \Rightarrow x = - 1
Putting the value of x in {x^{40}} + 3, we get the Ans. 4
Q.7) {67^{67}} + 67 is divided by 68 then remainder is?
Solution: \frac{{{{67}^{67}} + 67}}{{68}} = \frac{{{{(67)}^{67}}}}{{68}} + \frac{{67}}{{68}} 
                         = {( - 1)^{67}} - 1 = - 1 - 1 = - 2
                         = 68 - 2 = 66 Ans.

Number Series Formulae

a) Sum of first n natural numbers

 1 + 2 + 3 + 4........ + n = \frac{{n(n + 1)}}{2}
b) Sum of square of first n natural numbers
{1^2} + {2^2} + {3^2} + {4^2}........ + {n^2} = \frac{{n(n + 1)(2n + 1)}}{6}
c) Sum of cubes of first n natural numbers
{1^3} + {2^3} + {3^3} + {4^3}........ + {n^3} = {[\frac{{n(n + 1)}}{2}]^2}
d) Sum of first n odd numbers = {n^2}
e) Sum of first n even numbers = n(n + 1)

Trick to Find Square Root & Cube Root

a) If unit digit of a perfect number is 1 then unit digit of square root will be 1 or 9.
b) If unit digit of a perfect number is 4 then unit digit of square root will be 2 or 8.
c) If unit digit of a perfect number is 6 then unit digit of square root will be 4 or 6.
d) If unit digit of a perfect number is 5 then unit digit of square root will be 5 only.
e) If unit digit of a perfect number is 9 then unit digit of square root will be 3 or 7.

*Note: We never get square root of a number whose unit digit is 2, 3, 7 and 8.


                 

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