Multiples of Reciprocals with Maximum Period form Magic Square

Theorem
Let $p$ be a prime number whose reciprocal has a decimal expansion which has the maximum period, that is, $p - 1$.

Let the first $p - 1$ digits of $\dfrac 1 p$ be expressed as an integer $n$ (with one or more leading zeroes as it is presented).

Then the first $p - 1$ multiples of $n$, when listed in order of size, arrange themselves as a magic square such that:
 * the sum of the elements of each row
 * the sum of the elements in each column

are the same.