# Period of Reciprocal of 37 has Length 3

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## 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

Performing the calculation using long division:

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


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

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

$\blacksquare$