Reciprocal of 31

Theorem

The decimal expansion of the reciprocal of $31$ is as follows:

$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$

This sequence is A021035 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division: 

   0.03225806451612903...
  -----------------------
31)1.00000000000000000000
     93     155
     --     ---
      70      50
      62      31
      --      --
       80     190
       62     186
       --     ---
       180      40
       155      31
       ---      --
        250      90
        248      62
        ---      --
          200    280
          186    279
          ---    ---
           140     100
           124      93
           ---     ---
            160    ...
            155
            ---

$\blacksquare$

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