Many little divisors

Determine the largest integer \(n\) that is divisible by all positive integers (strictly) less than \(\dfrac{\sqrt{n}}{2}\).

As an example, for \(n = 36\), \(\frac{\sqrt{n}}{2} = 3\), and all positive integers less than it (\(1\) and \(2\)) divide \(n\), so it is one candidate. Find the largest one.

If there is no such largest integer (there's an arbitrarily large one), answer 0 instead.

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