by hemangsarkar, Sep 30, 2015, 10:20 AM
Q) Can we have a power of

such that if we permute its digits, we get another power of
?
SolutionThere can be at most 3 more candidates for any given power of 2.
We can see that it is true since after that we will be multiplying by

and that increases the number of digits by
.
Example:

are the only powers of

with

digits.
Also since we are only permuting the digits, the digit sum remains the same.
The digit sum modulo 9 is a useful tool from here.
It can be shown that the digit sum modulo 9 for

is the same as that of
.
The cycle is of length
.
This gives us a contradiction.
This post has been edited 1 time. Last edited by hemangsarkar, Sep 30, 2015, 10:20 AM