Let $F$ be a free abelian group, and let $\hat{F}_p$ denote its completion at a given prime $p$, why is it true that $$Tor^{\mathbb{Z}\lbrack\hat{F}_p/F\rbrack}_a(\mathbb{Z}, Tor^{\mathbb{Z}\lbrack F\rbrack}_b(\mathbb{Z}, \mathbb{F}_p))=0$$ for $a>0$.
In particular, I know that the fact that $\hat{F}_p/F$ is a uniquely $p$-divisible abelian group is important, but, somehow, I don't get how.
I also have a small question, if $A$ is an abelian group local away from $p$, is it true that $Tor_a^{\mathbb{Z}\lbrack A\rbrack}(\mathbb{Z},\mathbb{F}_p)=0$ for every interger $a$?