Highly divisible cycles in homological Atiyah-Hirzebruch spectral sequences

Let $F_*$ be a homology theory of finite type and $\{E^r_{s,t}(X)=\frac{Z^r_{s,t}(X)}{B^r_{s,t}(X)}; d^r_{s,t}\}$ denote the Atiyah-Hirzebruch spectral sequence for $F_*$ of a $CW$-complex $X$. In this paper we study elements of the maximal possible order in the groups $E^2_{n,0}(X)/Z^{r}_{n,0}(X)$.