6 - división rápido por 10n-1 o monodigit número
TRANSCRIPT
-
8/12/2019 6 - Divisin Rpido Por 10n-1 o Monodigit Nmero
1/5
101
Vitthal Jadhav
Pune, Maharashtra, India.
Abstract 101 .
1. B
1)Monodigit Number
.
.. ) 1111
) 222222
) 33
)
) .
,
* 111111111
D
* * 1111111111
D
*
D
*(101) ( . 10
1 )
D
-
8/12/2019 6 - Divisin Rpido Por 10n-1 o Monodigit Nmero
2/5
2)Numbers Polynomial Representation
.................. .........n mn n n mP D D D D D D D D D D D D += ,
+1 ,
. .
.................. .........
* * * ......... * * * * *
* ....... * *
n mn n n m
n n nn n n
m mmm
P D D D D D D D D D D D D
D D D D D D D D
D D D
b b b b b b b b
b b b
+ + + + + + + +
+ + + +
+
+ +
=
=
Pm
kk
k nD *b=
= (1)
A . +1
, .
P kk
k nD *b=
= (2)
2.
101 999999999.
..
n n
n n
nn
r
r
)
( )
y y(assuming x=10
( ) (x )
yx ( x )
yw assuming =w and x z
x( z )
w*( z z z z .....)
w* z
=
=
=
= =
= + + + + +
=
-
8/12/2019 6 - Divisin Rpido Por 10n-1 o Monodigit Nmero
3/5
r rr r
r m r mr
m m mm m mm m m
r
(substituting polynomial representation of w in base 'z' i.e. ( a *z )* z w = a *
= (a )z (a + a )z (a +a + a )z ..... -----------------(3)
b
= ==
=
+ + +
=
* -----------------------------
m m
m mm m m
mm m m m m
k k k
r (r m
4)
Comparing equation (3) & (4) we get
b a
b a a a b
b a a a a b
.
.
.
b a b -----------------(5)
z
+
=
=
= + = +
= + + = +
= +
A 999999
.
3.
1) 1234757777=?
7777= (7/9)(104 1)
, 123475 (7/9)* (1041) = 158753.5714285714(1041)
C (1) =10
=158753.5714285714 , =4
= =104
= / =15.87535714285714
=15*0+ 8753*1+5714*2+2857*3+1428*4+ ()
C () = rrr m
( a *z )
=
=0 ,0=15, 1=8753, 2=5714, 3=2857, 4=1428 .
-
8/12/2019 6 - Divisin Rpido Por 10n-1 o Monodigit Nmero
4/5
0= 0=15 ( (5) )
1= 1+ 0 =8753+15=8768
2= 2+ 1 =5714+8768=14482
3= 3+ 2 =2857+14482= 17339
4= 4+ 3 =1428+17339=18767
..
..
..
.
A (4)
.. (1) = rr m
( b *z )r
=
=0*0+ 1*1+ 2*2+ 3*3 +
=15*0+ 8768 *1+14482 *2+17339*3+187674+=15.8769 4483 7340 8767 ( =104)
1234757777 =15.8769 4483 7340 8767
)
*( )
**
( )
15.8753 5714285714 =
=
=
15. 53 514 25 142
+ 15 1442 133
15 . 1442 133 1 .
1 1 1
15. 443 340 .
-
8/12/2019 6 - Divisin Rpido Por 10n-1 o Monodigit Nmero
5/5
15. 53 514 25 142
+ 15 (1+442) 340
15 . 1442 340 .
+1 +1
15. 443 340
12345 15. 443 340 ..
2) 2367428142857= ?
.
= =
*
. =*( ) ( )
.
( )
=
=
=
=
=
.
( )
.
1. 51 000000 000000
+ 00001 52 012 52 012 ..
1 . 52 012 52 012 52 012
1 . 52 012 52 012 52 012 ..
2367428142857=16 . 572 012 572 012 572 012 .