Let $J_p$ be the additive group of $p$-adic integers and $Q_p$ be the additive group of $p$-adic numbers. I know that $J_p$ is residually finite.
Is $Q_p$ residually finite?
Definition
A group $G$ is said to be residually finite if the intersection of all subgroups of finite index is the trivial subgroup.
This is a divisible group and such a group cannot be residually finite.
Edit: for a positive integer $n$ let us consider the statement
$$S_n:\forall x\ \exists y: \underbrace{y+\cdots+y}_{n\text{ times}}=x.$$
An Abelian group $G$ being "divisible" means that $G$ satisfies this sentence for every $n$. Suppose that $G/H$ is a finite quotient of an Abelian group $G$. Then there exists a positive number $n$ such that for every $gH\in G/H$, $\underbrace{gH+\cdots+gH}_{n}=0$. On the other hand, every quotient of a divisible group must also satisfy all sentences $S_n$. This implies that a divisible group cannot have finite quotients (except the trivial one).