# 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 \(S\)?