37 is Second Number whose Period of Reciprocal is 3

Theorem
$37$ is the $2$nd positive integer the decimal expansion of whose reciprocal has a period of $3$:
 * $\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

Proof
Performing the calculation using long division:

0.027... 37)1.00000    74     --     260     259     ---       100        74       ---        ...

It can be determined by inspection of all smaller integers that this is indeed the $2$nd to have a period of $3$.