Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$?

[i]2018 CCA Math Bonanza Lightning Round #2.1[/i]
1. **Define the sets \( S \) and \( T \):**
   - \( S \) is the set of the first 2018 positive integers: \( S = \{1, 2, 3, \ldots, 2018\} \).
   - \( T \) is the set of all distinct numbers of the form \( ab \), where \( a \) and \( b \) are distinct members of \( S \).

2. **Understand the structure of \( T \):**
   - Each element in \( T \) is a product of two distinct elements from \( S \).
   - The smallest possible product in \( T \) is \( 1 \times 2 = 2 \).
   - The largest possible product in \( T \) is \( 2017 \times 2018 \).

3. **Determine the 2018th smallest element in \( T \):**
   - Since \( T \) contains products of distinct pairs from \( S \), we need to consider the sequence of products in increasing order.
   - The smallest products will be those involving the smallest integers in \( S \).

4. **Identify the pattern in the products:**
   - The smallest products are \( 1 \times 2, 1 \times 3, 1 \times 4, \ldots, 1 \times 2018 \).
   - Next, we have \( 2 \times 3, 2 \times 4, \ldots, 2 \times 2018 \).
   - Continue this pattern until we reach products involving larger integers.

5. **Count the number of products:**
   - For \( a = 1 \), there are 2017 products: \( 1 \times 2, 1 \times 3, \ldots, 1 \times 2018 \).
   - For \( a = 2 \), there are 2016 products: \( 2 \times 3, 2 \times 4, \ldots, 2 \times 2018 \).
   - Continue this pattern until \( a = 2017 \), which has 1 product: \( 2017 \times 2018 \).

6. **Sum the number of products:**
   - The total number of products is the sum of the first 2017 positive integers:
     \[
     \sum_{k=1}^{2017} k = \frac{2017 \times 2018}{2} = 2033136
     \]
   - Since we need the 2018th smallest product, we need to find the product that appears in the 2018th position in the sorted list of products.

7. **Identify the 2018th smallest product:**
   - The first 2017 products are all the products involving \( 1 \) (i.e., \( 1 \times 2, 1 \times 3, \ldots, 1 \times 2018 \)).
   - The 2018th product will be the smallest product involving \( 2 \) and another integer greater than \( 2 \).

8. **Calculate the 2018th product:**
   - The next smallest product after \( 1 \times 2018 \) is \( 2 \times 3 = 6 \).

The final answer is \( \boxed{6} \).