Call A1A2 y, a two-digit number, and call the rest of the numerator x.

17(y*10^k + x) = 12(100x + y),  where k = n-2

17y*10^k + 17x = 1200x + 12y

(17*10^k - 12)y = 1183x

x = (17*10^k - 12)y / 1183

The following program uses this equation to demonstrate that 6 is the smallest value of n that allows a solution:

   4    kill "shiftdg2.txt":open "shiftdg2.txt" for output as #2

   5    point 255

  10    for n=3 to 100:k=n-2

  20       for y=10 to 99

  30         x = (17*10^k - 12)*y // 1183

  40         if x=int(x) and (x>0 or y>0) then

  41           :print n,y,x:ct=ct+1

  42           :print #2, n,y,x:ct=ct+1

  43           :numf=y*10^k+x:denf=100*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

  65    close #2

  70    end

  

The manually enhanced output below shows n, y and x on the first line. Division and equal signs have been added manually to show the resulting division equality  

  

 6   13   1868 

 131868 / 186813 = 12/17 

 6   26   3736 

 263736 / 373626 = 12/17 

 6   39   5604 

 395604 / 560439 = 12/17 

 6   52   7472 

 527472 / 747252 = 12/17 

 6   65   9340 

 659340 / 934065 = 12/17 

 

 plus the spurious line of output where a "carry" interferes with the digits, as x becomes larger than four digits:

 

 6   78   11208 

 791208 / 1120878   12//17