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Shifting the digit (Posted on 2019-03-11) Difficulty: 3 of 5
A1A2A3...An-1An/A2A3...An-1AnA1 = 12/17

What is the smallest value of n such that there exists solutions to the above equation?

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer solution | Comment 1 of 3
Call A1 y, a single-digit number, and call the rest of the numerator (A2 ... A(n-1)) x.

17(y*10^k + x) = 12(10x + y),  where k = n-1

This can be evaluated to

x = y(17*10^k - 12) / 103

UBASIC allows us the precision to solve this:

(n in the program is the k used above)

   4    kill "shiftdig.txt":open "shiftdig.txt" for output as #2
   5    point 255
  10    for n=2 to 100
  20       for y=0 to 9
  30         x = y*(17*10^n-12)//103
  40         if x=int(x) and (x>0 or y>0) then
  41           :print n+1,y,x:ct=ct+1
  42           :print #2, n+1,y,x:ct=ct+1
  43           :numf=y*10^n+x:denf=10*x+y
  44           :print #2,numf,denf,numf//denf
  45           :print numf,denf,numf//denf
  49           :if ct>10 then cancel for:goto 55
  50       next y
  55       if ct>10 then cancel for:goto 70
  60    next n
  70    end
  
With n=34, which is the smallest value for n:  

 1165048543689320388349514563106796 / 1650485436893203883495145631067961 = 12/17 

 2330097087378640776699029126213592 / 3300970873786407766990291262135922 = 12/17 

 3495145631067961165048543689320388 / 4951456310679611650485436893203883 = 12/17 

 4660194174757281553398058252427184 / 6601941747572815533980582524271844 = 12/17 

 5825242718446601941747572815533980 / 8252427184466019417475728155339805 = 12/17 

 6990291262135922330097087378640776 / 9902912621359223300970873786407766 = 12/17 


The raw output:  
  
 34   1   165048543689320388349514563106796 
 1165048543689320388349514563106796   1650485436893203883495145631067961   12//17 
 34   2   330097087378640776699029126213592 
 2330097087378640776699029126213592   3300970873786407766990291262135922   12//17 
 34   3   495145631067961165048543689320388 
 3495145631067961165048543689320388   4951456310679611650485436893203883   12//17 
 34   4   660194174757281553398058252427184 
 4660194174757281553398058252427184   6601941747572815533980582524271844   12//17 
 34   5   825242718446601941747572815533980 
 5825242718446601941747572815533980   8252427184466019417475728155339805   12//17 
 34   6   990291262135922330097087378640776 
 6990291262135922330097087378640776   9902912621359223300970873786407766   12//17 


  Posted by Charlie on 2019-03-11 17:42:32
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