Problem 1284

To solve this problem, we need to find the maximum number of elements in a subset $X$ of $S = \{1, 2, \ldots, 100\}$ such that no two distinct elements in $X$ multiply to an element in $X$.

1. Understanding the Problem:
- We need to ensure that for any $a, b \in X$ (where $a \neq b$), the product $ab \notin X$.

2. Constructing a Candidate Set:
- Consider the set $X = \{10, 11, 12, \ldots, 100\}$. This set contains 91 elements.

3. Verifying the Condition:
- We need to check if for any $a, b \in X$ (where $a \neq b$), the product $ab \notin X$.
- For $a, b \in \{10, 11, 12, \ldots, 100\}$, the smallest possible product is $10 \times 11 = 110$, which is greater than 100.
- Therefore, no product of two distinct elements in $X$ will be in $X$.

4. Optimality:
- To determine if this is the maximum possible size, consider the following:
- If we include any number less than 10 in $X$, say $k$, then $k \times 10 \leq 100$, which would force $k \times 10$ to be in $X$, violating the condition.
- Thus, including any number less than 10 would reduce the size of $X$ while maintaining the condition.

5. Conclusion:
- The set $X = \{10, 11, 12, \ldots, 100\}$ is optimal because it maximizes the number of elements while satisfying the given condition.

The final answer is $\boxed{91}$.