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Carry on, Decimal (Posted on 2019-05-20) Difficulty: 3 of 5
Consider the following series written in form of a decimal

A = 1/9 + 1/99 + 1/999 + ... + 1/(10100 - 1)

Find the digit in the 71st place after the decimal point.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution solution | Comment 1 of 3
1/9 is .111111...
1/99 is .01010101...
1/999 is .001001001001...

In term t(n) every nth digit position is 1; the others are zero.

71 is prime and would directly receive a 1 from only t(1) and t(71), adding to 2.

The 72nd position would get a 1 from each of its 12 divisors and so contributes a carry into position 71, making it a 3.

73 is also prime and contributes nothing to positions to its left.

74 has only four divisors and so also contributes nothing.

75 has six divisors, again contributing no carries.

At this point position 71 is pretty well insulated by distance from any carries beyond this.

So my answer is 3.

Check:

0.122324243426244526264428344628264449244828266430364628484432246748264832
24664830543244483246445226691717732117391735534523354117372133771721334433
55211757411341391573252755175933271737373577231378212132403258441432443240
36208220167620343628145456363626327716405224363424187260126416147646263420
36724632283920262313041542264252264314424714303936621920701534623122503518
46391666432022237244234022451273153386421623393926301245553366261663352376
18323125332282126923176224362535594230326639153018370735192230264322376228
36321913924625493525226912293313494232531946621822852219265714283535922435
27236326291643441965431263141862453263491522275652141956253963142746281265
68332549129614422623262741140231323422415227154541632234242348225537128938
18652122493117263812721423924322231845524836712413253932532144423756242518
26343298141495231530442962342223484629371594221852282248413446571235231952
20258334316251365424134224325331212248423821432277152814371617393117236241
236 

The above has more places than we need, but is useful for accuracy comparison. (It was done with point 200 in the below program).

    4   kill "CARRYON.TXT"
    5   open "CARRYON.TXT" for output as #2
   10   point 20
   20   A=0
   30   for I=1 to 100
   40     A=A+1/(10^I-1)
   50   next
   60   print A
   70   print #2,A
   80   close #2

which finds a 3 in the 71st position after the decimal point.

0.122324243426244526264428344628264449244828266430364628484432246748264832246648305432444832464431 
 
                                                                      ^
                                                                        |
                                                                                
                                                                                 

Edited on May 20, 2019, 7:26 pm
  Posted by Charlie on 2019-05-20 19:24:02

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