Hi. I'm trying to settle the following problem but having difficult making further progress. Any hints or suggestions would be really appreciated. Thank you~
Does there exist a positive integer which is a power of 2, such that we can obtain another power of 2 by rearranging it's digits?
So far I only have obvious approach, that is the sum of digits of 2^a and 2^b must match if they're equal by rearranging digits of one of them. In this approach we have 2^a = 2^b (mod 9) so tha a = b (mod 6) since 2^(|a-b|) = 1 (mod 9) and 6 is the order of 2 modulo 9. This takes care of the case when the number of digits of 2^a and 2^b are equal and furthermore reveals that one of 2^a and 2^b must contain a digit 0.