Power Cycles

A spreadsheet is a good tool for this puzzle. Can you explain this phenomenon?

Source: Original. Based upon a puzzle from reader Jimmy Chng Gim Hong.

Solutions were received from: Philippe Fondanaiche, Jospeh DeVincentis, John Hewson, Claudio Baiocchi.

Claudio wrote a nice HTML-based script to calculate the results for different bases.

I was most intrigued by the uniformity in the periods regardless of the base. And how the first power of a period was always less than or equal to the number of digits.

Joseph's table of results. 
    First power that repeats     Length of repeating cycle

      number of last digits        number of last digits
base   1   2   3    4    5          1   2   3    4    5
  2    5  22  103  504  2505        4  20  100  500  2500
  3    5  21  101  501  5001        4  20  100  500  5000
  4    3  11   52  252  1253        2  10   50  250  1250
  5    2   3    5    8    13        1   1    2    4     8
  6    2   7   28  129   630        1   5   25  125   625
  7    5   5   21  101   501        4   4   20  100   500
  8    5  21  101  502  2502        4  20  100  500  2500
  9    3  11   51  251  2501        2  10   50  250  2500
 10    3   3    4    5     6        1   1    1    1     1
 11    3  11   51  501  5001        1  10   50  500  5000
 12    6  21  102  502  2503        4  20  100  500  2500
 13    6  21  101  501  5001        4  20  100  500  5000
 14    4  12   53  254  1255        2  10   50  250  1250
 15    3   4    5    6     7        1   2    2    2     2
 16    3   6   26  126   627        1   5   25  125   625
 17    6  21  101  501  2501        4  20  100  500  2500
 18    6   6   23  104   505        4   4   20  100   500
 19    4  11   51  501  5001        2  10   50  500  5000
 20    3   3    4    5     6        1   1    1    1     1
[To obtain the first power of each cycle, subtract the "length of cycle" from the "first to repeat". In each case, the first power is always less than or equal to N. I found this surprisingly regular. And there exists a cycle length for which ALL bases cycle since any smaller cycle-lengths are factors of the larger ones. (i.e. for N=5, all bases cycle with a length of 5000.) - KD]
John Hewson included a little bit of analysis to explain the similarities in the periods among the different bases:

The puzzle is equivalent to evaluating the powers k^i (mod 10^N). There is a result of Euler that if g(m) is the number of numbers less than m which are relatively prime to m then x^g(m) =1 (mod m) provided that x is relatively prime to m. Also if x^h = 1 (mod m) then h divides g(m). Now g(100) = 40, g(1000) = 400, and g(10000) = 4000 and for values of k not divisible by 2 or 5 the cycle lengths shown in the table do indeed divide the corresponding value of g(10^N). It appears that the cycle lengths for other values of k have the same property. Not every value of k belongs to a cycle.

To illustrate, consider the case N = 2. the values of k: 2, 5, 6, 10, 14, 15, and 18, within the listed values, do not belong to a cycle. For example the cycle with k = 2 begins (and ends) at 4 and does not include 2, and the cycle with k = 15 consists of the 2 numbers 25 and 75. The cycles for k = 2 and k = 12 consist of identical sets of numbers but in a different order, and there are other overlapping cycles. The cases N = 3 and N = 4 have similar properties.

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