(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.

10   for B=4 to 500
15   Good=0
20   for N=0 to 999
30    X=(4*B^N-2)//(B-2)
35    if X=int(X) then Good=1:cancel for:goto 50
40   next
50   if Good=0 then print B;
60   next B

finds, until it overflows, the following bases that do not work:

 6  10  14  18  22  26  30  34  38  42  46  50  54  58  62  66  70  74  78  82
86  90  94  98  102  106  110  114  118  122  126  130  134  138  142  146  150
 154  158  162  166  170  174  178  182  186  190  194  198  202  206  210  214
 218  222  226  230  234  238  242  246  250  254  258  262  266  270  274  278
 282  286  290  294  298  302  306  310  314  318  322  326  330  334  338  342
 346  350  354  358  362  366  370  374  378  382  386  390  394  398
Overflow in 30

The bases that do not work are those that equal 2 mod 4. Any other base will have a solution.

Posted by Charlie on 2011-08-21 14:12:13