Let $N$ be a positive integer. There are $N$ tasks, numbered $1, 2, 3, \ldots, N$, to be completed. Each task takes one minute to complete and the tasks must be completed subjected to the following conditions:
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[*] Any number of tasks can be performed at the same time.
[*] For any positive integer $k$, task $k$ begins immediately after all tasks whose numbers are divisors of $k$, not including $k$ itself, are completed.
[*] Task 1 is the first task to begin, and it begins by itself.
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Suppose $N = 2017$. How many minutes does it take for all of the tasks to complete? Which tasks are the last ones to complete?
1. **Define the height of an integer \( n \):**
   The height \( h(n) \) of an integer \( n \) is defined as the sum of the exponents in its prime factorization. For example, if \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \), then \( h(n) = e_1 + e_2 + \cdots + e_k \).

2. **Determine the maximum height for \( N = 2017 \):**
   Since 2017 is a prime number, its prime factorization is simply \( 2017 = 2017^1 \). Therefore, the height \( h(2017) = 1 \).

3. **Generalize the task completion process:**
   Tasks are completed in increasing order of their heights. Task 1, which has height 0, is completed first. Tasks with height 1 are completed next, and so on. The total time to complete all tasks is \( 1 + \max_{1 \leq i \leq 2017} h(i) \).

4. **Calculate the maximum height for all integers from 1 to 2017:**
   - For \( n = 1 \), \( h(1) = 0 \).
   - For any prime number \( p \), \( h(p) = 1 \).
   - For composite numbers, the height is the sum of the exponents in their prime factorization.

   Since 2017 is a prime number, the maximum height among all numbers from 1 to 2017 is 1.

5. **Determine the total time to complete all tasks:**
   The total time is \( 1 + \max_{1 \leq i \leq 2017} h(i) = 1 + 1 = 2 \) minutes.

6. **Identify the tasks that are completed last:**
   The tasks with the maximum height are the last to be completed. Since the maximum height is 1, the tasks that are completed last are the prime numbers up to 2017.

The final answer is \( \boxed{ 2 } \) minutes. The tasks that are completed last are the prime numbers up to 2017.