(I) A certain tridecimal (base 13) positive integer starts with the digit 2. Moving the 2 from the beginning of the number to its end doubles it.

What is the minimum value of this number?

(II) Determine the general form of a positive integer n such that there does not exist any base-n positive integer, starting with the digit 2, that doubles itself when the digit 2 is shifted from the beginning of the number to its end.

In an (n+1)-digit number:

2*(2*13^n + x) = 13*x + 2

4*13^n + 2*x = 13*x + 2

11*x = 4*13^n - 2

x = (4*13^n - 2) / 11

for some values of n:

 n         x
 0       2/11
 1       50/11
 2       674/11
 3       8786/11
 4       114242/11
 5       1485170/11
 6       19307234/11
 7       250994066/11
 8       3262922882/11
 9       3856181590
 10      551433967394/11
 11      7168641576146/11
 12      93192340489922/11
 13      1211500426369010/11
 14      15749505542797154/11
 15      204743572056363026/11
 16      2661666436732719362/11
 17      34601663677525351730/11
 18      449821627807829572514/11
 19      531607378318344040246
 20      76019855099523197755202/11
 21      988258116293801570817650/11
 

so the first n that works (x is an integer) is 9.
 
The 3,856,181,590 must be converted to base-13 before we can put a 2 in front of it and call part I solved. That conversion comes out to 495BA8371, so the answer to part I is:
 
 2495BA8371
 

Posted by Charlie on 2011-08-21 13:47:32