(* {{4,8,16},{2,3,9}}; 0; bytheory;

Anything with divisors Phi_4, 8,16 must tile with the C-M universal tile coming from Phi_2(x)=1+x, i.e. the set {0,1}. So, modulo 16 the only possibility is T={0,2,4,6,8,10,12,14}. Modulo 144 the only possible sets T' are of the form:  the elements of T (except for 0) are shifted by multiples of 16. There are 9^7 such sets. They could be checked within 2-3 days, but there is no need. Any set T' is composed of even numbers only. Now, there are 
two cases:
1. T' is periodic; we are done.

2. T' in non-periodic. Then a tiling comlement B of T' must be composed of some even numbers B1 
(tiling all the even numbers mod 144 with T') and some odd numbers B2. Both B1 and B2 induce a tiling of the factor group Z_72: (B1)/2+T'/2=Z_72 (the subgroup of all the even numbers mod 144). But we know from the case of tilings of Z_72
(which are all classified) that in this case (B1)/2 must be periodic by 24. And similarly B2/2 is also periodic by 24. Therefore B1, B2 and henceforth B is periodic by 48.

We are done.
*)