The torsion submodule of $\prod \mathbb{Z}_p$ is not a direct summand of $\prod \mathbb{Z}_p$

Let $G=\prod_p\Bbb Z_p$ and $H=\bigoplus_p\Bbb Z_p$ and regard $H$ as a subgroup of $G$. You want to know if $H$ is a direct factor of $G$. If it was there would be a subgroup $K$ of $G$ isomorphic to $G/H$. There isn't. You know that $G/H$ is divisible, but there are no nonzero elements $x\in G$ having the property that $x=p y$ for some $y\in G$ for all primes $p$.

Ask yourself this question: if $x \in \prod_p \mathbb{Z}_p$ is such that $x \in n \prod_p \mathbb{Z}_p$ for each $n \in \mathbb{Z} \setminus \{0\}$, then what can we say about $x$?

You should be able to conclude that there is no embedding from $\frac{\prod_p\mathbb{Z}_p}{\oplus_p \mathbb{Z}_p}$ to $\prod_p \mathbb{Z}_p$