= Solution
Apply bipartite <superadditivity of quantum relative entropy> first to system $1$ and systems $2,\ldots,n$, and then repeat on the remaining joint state. <Mathematical induction> gives
$$
\boxed{
D(\rho_{1\ldots n}\|
\sigma_1\otimes\cdots\otimes\sigma_n)
\geq\sum_{j=1}^nD(\rho_j\|\sigma_j)}.
$$
This is <multipartite superadditivity of quantum relative entropy>. Repeated application of bipartite additivity similarly gives
$$
\boxed{
D(\rho_1\otimes\cdots\otimes\rho_n\|
\sigma_1\otimes\cdots\otimes\sigma_n)
=\sum_{j=1}^nD(\rho_j\|\sigma_j)}.
$$
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