Each digit of base the base-16 representation of a number corresponds exactly to 4 digits in the base-2 representation, which allows for trivial conversion between these bases.
Other combinations of bases are less straightforward, but still permit short-cutting the generic remainder-and-divide algorithm for converting bases. E.g., 3 digits of base-16 correspond to 4 digits of base 8, and 3 digits of base-256 correspond to 4 digits of base-64 (hence the motivation for the base-64 encoding standard for binary data).
These sorts of relations are all fairly easily discoverable by comparing them to base-2 as an intermediary. But is there a more generic method of discovering whether or not a digit ratio exists between two radices, and if so, what it is?