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