<s>[INST] Question on The multiplicative magic square Consider this magic square:

$$\begin{array} {|r|r|}\hline a & b & c \\ \hline d & e & f \\ \hline j & h & i \\ \hline  \end{array} $$
Where $a,b,c,d,e,f,j,h,i\in \mathbb N^*$ and the $\gcd$ of all nine elements is $1$.
This magic square is A multiplicative one such that the product in each row, column and diagonal is equal to $P$

The first question: Prove that $e^3=P$
The second question: construct a multiplicative magic square with the divisors of $100$
My attempt:
It’s not actually an attempt, it is just an observation, since the $\gcd(a,b,c,d,e,f,j,h,i)=1$ And $$aei=cej=beh=def=P$$
$e\in\mathbb N^* \implies e\geq1$, so we can divde by it:
$$ai=cj=bh=df$$
Wich mean that $a\mid cj,bh,df$ And $i\mid cj,bh,df$ and so on..., but what’s next?
 [/INST] You can start with any additive magic square. The following are the simplest
\begin{array}{c}
   \begin{pmatrix}
       0 & 2 & 1 \\
       2 & 1 & 0 \\
       1 & 0 & 2 \\
   \end{pmatrix} &
   \begin{pmatrix}
       1 & 0 & 2 \\
       2 & 1 & 0 \\
       0 & 2 & 1 \\
   \end{pmatrix} &
   \begin{pmatrix}
       1 & 2 & 0 \\
       0 & 1 & 2 \\
       2 & 0 & 1 \\
   \end{pmatrix} &
   \begin{pmatrix}
       2 & 0 & 1 \\
       0 & 1 & 2 \\
       1 & 2 & 0 \\
   \end{pmatrix} \\
\end{array}
Pick one or more and associate each to an array of some chosen base raised to those powers.
For example.
\begin{array}{c}
   \begin{pmatrix}
       2^0 & 2^2 & 2^1 \\
       2^2 & 2^1 & 2^0 \\
       2^1 & 2^0 & 2^2 \\
   \end{pmatrix} &
   \begin{pmatrix}
       3^1 & 3^0 & 3^2 \\
       3^2 & 3^1 & 3^0 \\
       3^0 & 3^2 & 3^1 \\
   \end{pmatrix} &
   \begin{pmatrix}
       5^1 & 5^2 & 5^0 \\
       5^0 & 5^1 & 5^2 \\
       5^2 & 5^0 & 5^1 \\
   \end{pmatrix}
\end{array}
Simplify.
\begin{array}{c}
   \begin{pmatrix}
       1 & 4 & 2 \\
       4 & 2 & 1 \\
       2 & 1 & 4 \\
   \end{pmatrix} &
   \begin{pmatrix}
       3 & 1 & 9 \\
       9 & 3 & 1 \\
       1 & 9 & 3 \\
   \end{pmatrix} &
   \begin{pmatrix}
       5 & 25 & 1 \\
       1 & 5 & 25 \\
       25 & 1 & 5 \\
   \end{pmatrix}
\end{array}
Create one array by multiplying pointwise.
\begin{pmatrix}
       15 & 100 & 18 \\
       36 & 30 & 25 \\
       50 & 9 & 60 \\
   \end{pmatrix}
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