Period of Reciprocal of 7 is of Maximal Length

Theorem
$7$ is the smallest integer $n$ the decimal expansion of whose reciprocal has the maximum period $n - 1$, that is: $6$:
 * $\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$


 * ReciprocalOf7Cycle.png

Proof
Performing the calculation using long division:

0.1428571 -- 7)1.0000000   7  ---    30    28    --     20     14     --      60      56      --       40       35       --        50        49        --         10          7         --         .....

The reciprocals of $1$, $2$, $4$ and $5$ do not recur:

while those of $3$ and $6$ do recur, but with the non-maximum period of $1$: