How many positive integers completely divide p^3 but do not completely divide p?

To resolve the question, it's important to analyze the properties of the integer ( p ) when raised to different powers. Let’s denote ( p ) as a prime number to simplify our analysis.

When dealing with prime factors, the number of positive divisors (or "divisors") of a number can generally be determined using its prime factorization. For any integer ( n = p^k ) (where ( p ) is prime and ( k ) is a positive integer), the total number of positive divisors ( d(n) ) is given by the formula ( k + 1 ).

In the case of ( p^3 ), the prime factorization shows us that it can be expressed as ( p^3 ). This means it has ( 3 + 1 = 4 ) positive divisors: ( 1, p, p^2, ) and ( p^3 ).

Now, we need to identify how many of these divisors completely divide ( p^3 ) but do not completely divide ( p ). The divisors of ( p ) are simply ( 1 ) and ( p ) (which are ( 2