37 is Second Number whose Period of Reciprocal is 3

Theorem

$37$ is the $2$nd positive integer (after $27$) the decimal expansion of whose reciprocal has a period of $3$:

$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

Proof

From Reciprocal of $37$:

$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

Counting the digits, it is seen that this has a period of recurrence of $3$.

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

$\blacksquare$