  Let $G$ be a bipartite graph with bipartition $(X,Y)$, let $k$ be a positive
integer, and let $f:V(G)\rightarrow Z_k$ be a mapping with $\sum_{v\in X}f(v)
\stackrel{k}{\equiv}\sum_{v\in Y}f(v)$. In this paper, we show that if $G$ is
$(3k-3)$-edge-connected, then it admits a factor $H$ such that for each vertex
$v$, $d_H(v)\stackrel{k}{\equiv} f(v)$, and
$$\lfloor\frac{d_G(v)}{2}\rfloor-(k-1)\le d_{H}(v)\le
\lceil\frac{d_G(v)}{2}\rceil+k-1.$$ Next, we generalize this result to general
graphs and derive a sufficient degree condition for a highly edge-connected
general graph $G$ to have a factor $H$ such that for each vertex $v$,
$d_H(v)\in \{f(v),f(v)+k\}$. Finally, we show that every
$(4k-1)$-edge-connected essentially $(6k-7)$-edge-connected graph admits a
bipartite factor whose degrees are positive and divisible by $k$.
