27 is Smallest Number whose Period of Reciprocal is 3

Theorem
$27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$:
 * $\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$

Proof
Performing the calculation using long division:

0.037... 27)1.00000    81     --     190     189     ---       100        81       ---        ...

This is because $999 = 27 \times 37$.

It can be determined by inspection of all smaller integers that this is indeed the smallest to have a period of $3$.