Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.
1. **Identify the first term and its properties:**
   The first term of the arithmetic progression is given as \(16\). We note that:
   \[
   16 = 2^4
   \]
   The number of positive divisors of \(16\) is:
   \[
   4 + 1 = 5
   \]
   This is because the number of divisors of \(2^4\) is given by \(4 + 1\).

2. **Determine the common difference:**
   We need to find the smallest common difference \(d\) such that every term in the sequence \(16, 16+d, 16+2d, \ldots\) has a number of divisors divisible by \(5\).

3. **Formulate the general term:**
   The general term of the arithmetic progression can be written as:
   \[
   a_n = 16 + nd
   \]
   We need \(a_n\) to have a number of divisors divisible by \(5\).

4. **Analyze the number of divisors:**
   For a number \(a_n = 16 + nd\) to have a number of divisors divisible by \(5\), it must be of the form \(p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}\) where the product \((e_1 + 1)(e_2 + 1) \cdots (e_k + 1)\) is divisible by \(5\).

5. **Choose a suitable common difference:**
   We start by considering the next number after \(16\) that has exactly \(5\) divisors. The smallest such number is \(48\):
   \[
   48 = 2^4 \times 3
   \]
   The number of divisors of \(48\) is:
   \[
   (4+1)(1+1) = 5 \times 2 = 10
   \]
   Since \(48\) has \(10\) divisors, which is divisible by \(5\), we consider the common difference:
   \[
   d = 48 - 16 = 32
   \]

6. **Verify the sequence:**
   We need to check if the sequence with common difference \(32\) satisfies the condition. The general term is:
   \[
   a_n = 16 + 32n = 2^4 (1 + 2n)
   \]
   The number of divisors of \(a_n\) is:
   \[
   (4+1)(e_2 + 1)
   \]
   where \(1 + 2n\) is factored into primes as \(p_2^{e_2}\). Since \(4+1 = 5\), the number of divisors is always a multiple of \(5\).

7. **Conclusion:**
   The smallest possible common difference that satisfies the condition is \(32\).

The final answer is \(\boxed{32}\)