Question on The multiplicative magic square

Question

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?

Answer 1

$$e^3=\frac{aei\cdot beh\cdot ceg}{abc\cdot ghi}=\frac{P^3}{P^2}=P$$

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The knight move strategy seems to work just as in the additive case:

$$\begin{matrix}50 &1&20\\4&10& 25\\5&100 &2\end{matrix}$$

Answer 2

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}