# Another mean question

Number Theory Level pending

Let $$S$$ be a set of (distinct!) prime numbers, and let $$\langle S\rangle_G$$ be the geometric mean of $$S$$:

$\langle S\rangle_G = \sqrt[\#S]{\prod S}.$

Suppose there exist a prime number $$p$$ and a positive integer $$n$$ such that $$\langle S \rangle_G^n = p$$. What is the greatest possible number of elements in $$S$$?

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