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