# Period of Reciprocal of 27 is Smallest with Length 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 $3$.

$\blacksquare$