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:
 * $\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$

From Reciprocal of 131:
 * $\dfrac 1 {131} = 0 \cdotp \dot 00763 \, 35877 \, 86259 \, 54198 \, 47328 \, 24427 \, 48091 \, 60305 \, 34351 \, 14503 \, 81679 \, 38931 \, 29770 \, 99236 \, 64122 \, 13740 \, 45801 \, 52671 \, 75572 \, 51908 \, 39694 \, 65648 \, 85496 \, 18320 \, 61068 \, 7022 \dot 9$