Disjoint Covering systems

Question

Can their exist a disjoint covering system of Z+, made up of an infinite quantity of equivalence classes?

Answer 1

Yes, for example the system given by {1(mod 2), 2(mod 4), 4(mod 8), 8(mod 16), ...}.

Of course one would have to justify why this works. First notice that any positive integer can be written uniquely as a power of 2 times an odd number. Symbolically we say that any n in Z+ is equal to 2^k (2m+1) for some nonnegative integers k and m. This means that n=2^{k+1}m+2^k and therefore it is 2^k(mod 2^{k+1}), so it belongs to one and only one of the equivalence classes in our covering system.