= Solution
For a locally compact group $G$ on which multiplication by $n$ has finite kernel and cokernel, compare a <Haar measure> with its pushforward under $[n]$. On the one-dimensional $p$-adic Lie group $E(\mathbb Q_p)$, the derivative of $[n]$ at the identity is $n$, hence
$$
\frac{\#E(\mathbb Q_p)/nE(\mathbb Q_p)}{\#E(\mathbb Q_p)[n]}
=|n|_p^{-1}=p^{v_p(n)}.
$$
This can also be read directly from the successive quotients in the <filtration of elliptic-curve points over a local field>; factors prime to $p$ act invertibly on a sufficiently small formal-group neighbourhood.
The real Lie group $E(\mathbb R)$ has one or two circle components. On its identity component, $[n]$ has degree $n$; the component-group kernel and cokernel have the same order. Therefore
$$
\frac{\#E(\mathbb R)/nE(\mathbb R)}{\#E(\mathbb R)[n]}=\frac1n.
$$
Only primes dividing $n$ contribute to the finite-place product, and unique factorization gives
$$
\frac1n\prod_p p^{v_p(n)}=1.
$$
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