Smallest Multiplicative Magic Square is of Order 3

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

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.