In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ to $\begin{pmatrix} 1 & 20 \\ 3 & 40 \end{pmatrix}$ and then to $\begin{pmatrix} 1 & 20 \\ 6 & 80 \end{pmatrix}$.

A tourist attraction is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$, or $16$. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?

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