I have a **simply-connected** CW-complex $F$ of **finite-type**, and I know that the imprimitivity of its particular integral homology is **divisible** by an odd prime $p$; that is,
$$ \forall n,\exists \delta, \forall x : H_n(F,\mathbb{Z}), \nabla x = x\otimes 1 + 1 \otimes x + p (\delta x) $$
for a very nice delta.

Can I already conclude that an equivalent complex can be chosen with cellular attaching maps also divisible by $p$? Or do I need to know something more about the complex?