Number System (IDENTITIES)

IDENTITIES

1. Basic Formulas :

• (a + b)² = a² + b² + 2 ab

• (a - b)² = a² + b² - 2 ab

• (a + b)² + (a - b)² = 2 ( a² + b² )

• (a + b)² - (a - b)² = 4 ab

• a² - b² = (a + b) (a - b)

• (a + b + c)² = a² + b² + c² + 2 ( ab + bc + ca)

• (a + b)³ = a³ + b³ + 3ab (a + b) = a³ + b³ + 3a²b + 3ab²

• (a - b)³ = a³ - b³ - 3ab (a - b) = a³ - b³ - 3a²b + 3ab²

• a³ + b³ = ( a + b ) ( a² + b² - ab )

• a³ - b³ = ( a - b ) ( a² + b² + ab )

• (a³ + b³+ c³ - 3 abc)  = (a + b+ c) (a² + b² + c² - ab - bc - ca)

In the above identity ;

CASE 1 :  if (a + b+ c) = 0 

(a³ + b³+ c³ - 3 abc)  = 0

Hence, a³ + b³+ c³ = 3abc

CASE 2 : (a³ + b³+ c³ - 3 abc)  = ½ { 2(a + b+ c) (a² + b² + c² - ab - bc - ca)} 

Or,  (a³ + b³+ c³ - 3 abc) = ½ {(a + b+ c) (2a² + 2b² + 2c² - 2ab - 2bc - 2ca)

Or,  (a³ + b³+ c³ - 3 abc) = ½ {(a + b+ c) ( a - b )² ( b - c )² ( c - a )² }

2. Division Algorithm : 

When a number is divisible by another number then;

                  DIVIDEND = ( DIVISOR * QUOTIENT) + REMAINDER

3. Important Facts : 

• If a number N is divisible by  two co-prime numbers a and b then N is divisible by A*B.

• ( 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.

4. Series :

• ( 1 + 2 + 3 + 4 + ………….n) = ½ {n (n+1 )}

• ( 1² + 2² + 3² + 4² + ………….n²) = ⅙ { n ( n + 1) ( 2n + 1)}

• ( 1³ + 2³ + 3³ + 4³ + ………….n³) =  [½ {n (n+1 )}]²

• (1 + 3 + 5 + 7 + ................... ) = n² ( n = no. of terms)

• ( 2 + 4 + 6 + 8 + ....................) = n (n + 1 )

PRACTICE MATERIAL :

Q1. What is the sum of 50 natural numbers ?

1. 1275

2. 2550

3. 1425

4. None of these

Sol : Sum of 50 Natural Numbers = ½ ( 50 * 51 ) = 1275

Q 2. The smallest three digit prime number is :

1. 101

2. 103

3. 109

4. 113

Sol : 101 is the smallest prime number.

Q3. Calculate

629 x 629 + 371 x  371  - 629 x 371	=
629 x 629 x 629 + 371 x 371 x  371

1. 1/1000

2. 1/ 258

3. 258 / 500

4. 1000

Sol : Using the identity ; a³ + b³ = ( a + b ) ( a² + b² - ab )

Q 4. Calculate : 93 x 93 + 73 x 73 - 2 x 93 x 73

1. 400

2. 300

3. 200

4. 100

Sol : Using the identity ; (a - b)² = a² + b² - 2 ab

Q 5. (112 + 122 + 132 + ... + 202) = ?

1. 385

2. 2485

3. 2870

4. 3255

Sol : Using the Formula ; ( 1² + 2² + 3² + 4² + ………….n²) = ⅙ { n ( n + 1) ( 2n + 1)}

Q6.  The largest 4 digit number exactly divisible by 88 is:

1. 9944

2. 9768

3. 9988

4. 8888

Sol : Divisibility of 88 = A number is divisible by 88 if it is divisible by 8 and 11 both.

Q 7.  If the number 517#324 is completely  divisible by 3,  then the smallest whole number in the place of * is : 

1. 0

2. 1

3. 2

4. None of these

Sol : Sum of the digit should be divisible by 3 = 5 + 1 + 7 + # + 3 + 2 + 4 = 22 + #

# should be replaced with 2 , so as to make the sum divisible by 3.

Q 8. If the product 4864 x 9P2 is divisible by 12, then the value of P is :

1. 2

2. 5

3. 6

4. None of these

Sol : Divisibility of 12 = Number should be divisible by 3 & 4 both.

 4864 is divisible by 4

Hence 9P2 should be divisible by 3 = 9 + P + 2 = 11 + P 

Here P should be replaced with 1 

Q 9. What smallest number should be added to 4456 so that the sum is completely divisible by 6?

1. 4

2. 3

3. 2

4. 1

Sol : Divisibility of 6 = Number should be divisible by 2 & 3 both.

4456 is divisible by 2

Divisibility of 3 :  4 + 4 + 5 + 6  =19

19 + 2 = 21 ; Divisible by 2

Q 10.  476#$0 is divisible by both 3 and 11. The no - digits in the hundred’s and ten’s places are respectively :

1. 7 and 4

2. 7 and 5

3. 8 and 5

4. None of these

Sol : Divisibility of 3 : 4 + 7 + 6 + # + $ + 0 = (17 + # + $) must be divisible by 3

Divisibility of 11 : ( 7 + # + 0 ) - (4 + 6 + $ )  = ( 7 + # ) - (10 + $ ) = (# - $ - 3) must be divisible by 11

On solving we get, # = 8 and $ = 5

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