Let
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
be a prime number and let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
be a set of positive integers that satisfies the following conditions: (1) the set of prime divisors of the elements in
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
consists of
![p-1](/media/m/3/8/9/389a51fc4e0296668e79f23ead7e404f.png)
elements; (2) for any nonempty subset of
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
, the product of its elements is not a perfect
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
-th power. What is the largest possible number of elements in
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
?
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Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (1) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (2) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power. What is the largest possible number of elements in $A$ ?