To solve this problem, let’s analyze the conditions under which a number has exactly 9 divisors.

Key Insight:

A number $n$ has exactly 9 divisors if it can be expressed in a form that meets certain criteria based on the divisor function. Specifically:

1. If $n = p^8$ (where $p$ is a prime), then $n$ will have exactly 9 divisors, since the divisors are $1, p, p^2, \dots, p^8$.
2. If $n = p^2 \cdot q^2$ (where $p$ and $q$ are distinct primes), then $n$ will also have exactly 9 divisors.

Steps:

1. Identify 4-digit numbers in the form $p^8$:

   • For this form, $p^8$ should be a 4-digit number, meaning $1000 \leq p^8 \leq 9999$.
   • Solving this inequality, we find possible values of $p$ that satisfy this condition.

2. Identify 4-digit numbers in the form $p^2 \cdot q^2$:

   • Here, $p$ and $q$ must be distinct primes, and $p^2 \cdot q^2$ should be a 4-digit number (i.e., $1000 \leq p^2 \cdot q^2 \leq 9999$).

Let me carry out these calculations and list the valid numbers for each form.There are exactly 24 four-digit numbers that have exactly 9 divisors from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. These numbers are:

$1089, 1156, 1225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6561, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025$

Would you like further details on how the calculations were performed?

Here are some related questions for further exploration:

1. What are the properties of numbers with a fixed number of divisors?
2. How can we generalize the approach to find numbers with exactly $n$ divisors?
3. What is the significance of the prime powers in determining the number of divisors?
4. How does the divisor function behave under multiplication of distinct primes?
5. What would change if we were only interested in divisors less than or equal to 10?

Tip: Remember, the number of divisors of a number $n = p^a \times q^b$ (where $p$ and $q$ are primes) is given by $(a+1)(b+1)$. This can help quickly determine the number of divisors without explicitly listing them.