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