I want to prove that the positive powers of two, mod 10m, cycle with period 4\*5m-1. It's simple to prove that the powers of FIVE cycle with this period (2 is a primitive root mod powers of five), but how do you make the leap to powers of TEN? 


I'm sure it's something simple -- perhaps related to the Chinese Remainder Theorem -- but I don't see the connection yet.


Thanks for the help.



Cycle Length of the Positive Powers of Two Mod Powers of Ten
The answer I like best is based on the proof in the "physics forums" thread linked to in the comments above. I wrote about it in detail here: <http://www.exploringbinary.com/cycle-length-of-powers-of-two-mod-powers-of-ten/>


