The square

$\begin{tabular}{|c|c|c|} \hline 50 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 2 \\ \hline \end{tabular}$
is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of $g$?
$\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 136$
All the unknown entries can be expressed in terms of $b$.
Since $100e = beh = ceg = def$, it follows that $h = \frac{100}{b}, g = \frac{100}{c}$,
and $f = \frac{100}{d}$. Comparing rows $1$ and $3$ then gives
$50bc = 2 \cdot \frac{100}{b} \cdot \frac{100}{c}$,
from which $c = \frac{20}{b}$.
Comparing columns $1$ and $3$ gives
$50d \cdot \frac{100}{c}= 2c \cdot \frac{100}{d}$,
from which $d = \frac{c}{5} = \frac{4}{b}$.
Finally, $f = 25b, g = 5b$, and $e = 10$. All the entries are positive integers
if and only if $b = 1, 2,$ or $4$. The corresponding values for $g$ are $5, 10,$ and
$20$, and their sum is $\boxed{\mathbf{(C)}35}$.
Credit to Solution B goes to [http://billingswest.billings.k12.mt.us/math/AMC%201012/AMC%2012%20work%20sheets/2004%20AMC%2012B%20ws-15.pdf](https://artofproblemsolving.comhttp://billingswest.billings.k12.mt.us/math/AMC%201012/AMC%2012%20work%20sheets/2004%20AMC%2012B%20ws-15.pdf), a page with a play-by-play explanation of the solutions to this test's problems.