I have a **simply-connected** CW-complex of **finite-type**, and I know that the imprimitivity of its particular integral homology is **divisible** by $p$; that is,
$$ \nabla x = x\otimes 1 + 1 \otimes x + p (\delta x) $$
for a very nice delta.

Can I already conclude that the cellular attaching maps are also divisible by $p$? Or do I need to know something more about the complex?