ÖMO 2015 Problem 3

We consider the following operation applied to a positive integer: The integer is represented in an arbitrary base $b \ge 2$ , in which it has exactly two digits and in which both digits are different from $0$ . Then the two digits are swapped and the result in base $b$ is the new number.

Is it possible to transform every number $> 10$ to a number $\le 10$ with a series of such operations?

(Theresia Eisenkölbl)