Necessary and sufficient detection efficiency for the Mermin inequalities

We prove that the threshold detection efficiency for a loophole-free Bell experiment using an $n$-qubit Greenberger-Horne-Zeilinger state and the correlations appearing in the $n$-partite Mermin inequality is $n/(2n-2)$. If the detection efficiency is equal to or lower than this value, there are local hidden variable models that can simulate all the quantum predictions. If the detection efficiency is above this value, there is no local hidden variable model that can simulate all the quantum predictions.