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Erd\\H{o}s in 1938 studied the maximum size of a multiplicative Sidon subset of $\\{1,\\ldots, n\\}$, which was later determined up to the lower order term: $\\pi(n)+\\Theta(\\frac{n^{3/4}}{(\\log n)^{3/2}})$. We show that the number of multiplicative Sidon subsets of $\\{1,\\ldots, n\\}$ is $T(n)\\cdot 2^{\\Theta(\\frac{n^{3/4}}{(\\log n)^{3/2}})}$ for a certain function $T(n)\\approx 2^{1.815\\pi(n)}$ which we specify. 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