Reciprocals whose Decimal Expansion contain Equal Numbers of Digits from 0 to 9

Theorem
The following positive integers $p$ have reciprocals whose decimal expansions:
 * $(1): \quad$ have the maximum period, that is: $p - 1$
 * $(2): \quad$ have an equal number, $\dfrac {p - 1} {10}$, of each of the digits from $0$ to $9$:


 * $61$, $131$, $\ldots$

Proof
From Reciprocal of $61$:

From Reciprocal of $131$: