Find the largest positive integer $k$ such that there exists a set $A \subseteq \{1, 2, \ldots, 100\}$ with $k$ elements. For any $a, b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$.

Let's think step by step and output the final answer within \boxed{}.
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