So, this should work for all bases which have exactly one “2″ in their prime factorisation. Indeed, I don’t see why not, but I ought to do a demo, and the simplest choice is base 6 – senary, as it’s called. It does actually live up to its punnable name in terms of fraction representation, but that’s beyond the point.

(senary = binary)
1 = 1
10 = 110
100 = 100100
1000 = 11011000

As expected, all good! Also, the power of 2 in the prime factorisation can be used to tell you how many trailing zeros to expect. In my familiar base 2^2 * 3, for example, you get:

(dozenal = binary (comment))
10 ^ 0 = 1 (2 * 0 trailing zeros)
10 ^ 1 = 1100 (2 * 1 trailing zeros)
10 ^ 2 = 10010000 (2 * 2 trailing zeros)
10 ^ n has 2n trailing zeros.

This of course holds for odd bases (where the power of 2 is 0, and all powers are odd). But it only looks interesting for the “exactly one 2″ bases.