31 is Smallest Prime whose Reciprocal has Odd Period

Theorem
The decimal expansion of the reciprocal of $31$ has an odd period, that is: $15$:
 * $\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$

It is the smallest positive integer to have an odd period greater than $1$.

Proof
Performing the calculation using long division:

0.03225806451612903... --- 31)1.00000000000000000000     93     155     --     ---      70      50      62      31      --      --       80     190       62     186       --     ---       180      40       155      31       ---      --        250      90        248      62        ---      --          200    280          186    279          ---    ---           140     100           124      93           ---     ---            160    ...            155            ---