Smallest Multiplicative Magic Square is of Order 3

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Theorem

The order of the smallest multiplicative magic square is $3$, for example:

$\begin{array}{|c|c|c|} \hline 18 & 1 & 12 \\ \hline 4 & 6 & 9 \\ \hline 3 & 36 & 2 \\ \hline \end{array}$

Its magic constant is $216$.


Proof

Suppose there were an order $2$ multiplicative magic square $M$.

Let $a$ be the element of row $1$ and column $1$.

Let $a b$ be the magic constant of $M$.

Then $b$ is:

the element of row $1$ and column $2$, to make the product of row $1$ equal to $a b$
the element of row $2$ and column $1$, to make the product of column $1$ equal to $a b$

and so on.

But it is a principle of multiplicative magic squares that the elements are all distinct.

Hence no order $2$ multiplicative magic square can exist.

$\blacksquare$