Period of Reciprocal of 27 is Smallest with Length 3

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Theorem

$27$ is the smallest positive integer the decimal expansion of whose reciprocal has a period of $3$:

$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$

This sequence is A021027 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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 $3$.

$\blacksquare$


Sources