11 - algoritmo de novela por 'enésima raíz de número' usando expansión multinomial

8/12/2019 11 - Algoritmo de Novela Por 'Ensima Raz de Nmero' Usando Expansin Multinomial

1/9


2/9

2

3 3 2 2 2 2( 10 100 ) 3 *10 (3 3 )10 1000a b c a a b ab a c Y + + = + + + +

Now given cube = 3 2 2 2 23 *10 (3 3 )10 1000 150568768a a b ab a c Y + + + + =

3 2 2 2 23 *10 (3 3 )10 1000 149721291a a b ab a c Y + + + + =

2 2 2 2 33 *10 (3 3 )10 1000 149721291a b ab a c Y a+ + + = -------------(1)

As LHS is multiple of 10, So RHS 3149721291 a must be divisible by 10

3149721291 a must ends with 0 i.e. 3a must ends with 1.

Now there is only one integer i.e. 1 between 0 to 9 whose cube end with 1 , a=1

Substituting a=1 in above equation (1)

2 2 2 23 *10 (3 3 )10 1000 149721291 1 149721290a b ab a c Y + + + = =

2 2 23 (3 3 )10 100 14972129a b ab a c Y + + + = (Dividing both side by 10)

Similarly,

2 2 2(3 3 )10 100 14972129 3ab a c Y a b+ + =

LHS is multiple of 10 23a b =3b must ends with 9

2 2

2 2

2 2 2 2

(3 3 )10 100 14972129 3*3 14972120

(3 3 ) 10 1497212

10 1497212 (3 3 ) 1497212 (3*1*3 3*1 * )

10 1497185 3

LHS divisible by 10 3 must ends with 5 and 1 c 9 c =5

ab a c Y

ab a c Y

Y ab a c c

Y c

c

+ + = =

+ + =

= + = +

=

Cube root= cba= 531


3/9

3

In short,

( )3

3

1) ends with 1 End digit of 149721291

a 1 1 ,Subtracting it from 149721291 and 149721291

eliminating last digit i.e. 149721

a

a = =

290 14972129 1

---------------

=

( )2 149721290

2) 3 3 ends with 9 End digit of 14972129

b

a b b=

=2

3 3 9 ,Subtracting it from 14972129 and 14972129

eliminating last digit i.e. 14972120 1497212 9

a b =

=

-----------------

( )2 2

2 2

1497212

3) 3 3 27 3 ends with 2 End digit of 1497212 42

3 ends with 5 c 5 3 3 42 . -----

ab a c c

c ab a c

+ = +

= + = ------------

149717

Cube Root(150568768) 531cba = =

Applicability

Above method is convenient to extract nth

root of perfect nth

power that

satisfies

1(10, * ) 1

nGCD n U =

th(10, ) 1 and GCD(10, U) =1 Where U= Unit digit of n root .

i.e. Unit digits of " n and n root " must be relative prime to 10 (base).

(Why ? T

th

GCD n =

hink ).

2) FORWARD APPROACHExample 1) ( )

1th4(20352513413376) ? Given number is perfect 4 power=

Steps

1) Arrange given perfect nthpower into n digit groupfrom right to left.Here n = 4.

20 3525 1241 3376

No. of groups = 4 No. of digit in root = 4

Let root = 1000 100 10a b c d + + +


4/9

4

Why This Works ?

th

4

( ( ) has only one real positive root -According to Decarte's rule )

(Deca

Let N= 20 3525 1241 3376

x is 4 root of N

( ) = x 20 3525 1241 3376

f x

f x

4 4

4 4 4 12 12 4 12

rtes Rule- Maximum no. of positive root of

( ) No. of sign (coefficient sign) changes in ( ) )

2000 3000

2000

2 20 3 2 *10 20*10 3 *10

f x f x

N

=

11 - algoritmo de novela por 'enésima raíz de número' usando expansión multinomial

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