Find all positive integers $n$ with the following property: There are only a finite number of positive multiples of $n$ that have exactly $n$ positive divisors.
1. **Claim**: The positive integer \( n \) has the property that there are only a finite number of positive multiples of \( n \) that have exactly \( n \) positive divisors if and only if \( n \) is squarefree or \( n = 4 \).

2. **Prime Factorization**: Let \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) be the prime factorization of \( n \).

3. **Case 1: \( n \) is not squarefree**:
    - If \( e_i \ge 2 \) for some odd prime \( p_i \), then \( p_i^{e_i-1} \ge e_i + 1 \) (since \( p_i \ge 3 \)).
    - For the other primes \( p_j \), \( p_j^{e_j} \ge e_j + 1 \).
    - Consider the number \( p_i^{p_i^{e_i-1}-1} \cdot \prod_{j \ne i} p_j^{p_j^{e_j}-1} \cdot q^{p_i-1} \), where \( q \) is an arbitrary prime different from \( p_i \).
    - This number has \( n \) divisors and is a multiple of \( n \), implying there are infinitely many such multiples.

4. **Case 2: \( p_1 = 2 \) and \( e_1 \ge 3 \)**:
    - Here, \( 2^{e_1-1} \ge e_1 + 1 \).
    - Using the same argument as above, we can construct infinitely many multiples of \( n \) with exactly \( n \) divisors.

5. **Case 3: \( n = 4p_1p_2 \cdots p_k \) for \( k \ge 1 \)**:
    - Consider the number \( 2^{p_1-1} p_1 p_2^{p_2-1} \cdots p_k^{p_k-1} q \), where \( q \) is a prime different from \( p_i \).
    - This number has \( n \) divisors and is a multiple of \( n \), implying there are infinitely many such multiples.

6. **Case 4: \( n \) is squarefree**:
    - Suppose \( n \) is squarefree with \( k \) prime divisors.
    - Any number with \( n \) divisors can have at most \( k \) distinct prime divisors since each prime contributes at least one factor to the number of divisors.
    - These primes must be the primes dividing \( n \), and there are only finitely many such combinations.

7. **Special Case: \( n = 4 \)**:
    - The result for \( n = 4 \) is trivial because the only multiples of 4 with exactly 4 divisors are of the form \( 2^2 \cdot p \), where \( p \) is a prime.

Conclusion:
\[
\boxed{n \text{ is squarefree or } n = 4}
\]