If $p$ is a prime number and $n$ is a positive integer, then the number of positive divisors of $p^n$ is:

Step 1 (Identify positive divisors): Since $p$ is prime, the only prime factor of $p^n$ is $p$. Any positive divisor of $p^n$ must be a power of $p$, namely $p^k$, where $k$ is an integer.

Step 2 (Determine boundary conditions on exponent): For $p^k$ to divide $p^n$, the exponent $k$ must satisfy:

Step 3 (Count the divisors): The possible integer values for $k$ are:

Counting the elements in this sequence:

Step 4 (Conclusion): The prime power $p^n$ has exactly $n+1$ positive divisors. This corresponds to option index 1.