Reciprocal of 67

Theorem

The decimal expansion of the reciprocal of $67$ is as follows:

$\dfrac 1 {67} = 0 \cdotp \dot 01492 \, 53731 \, 34328 \, 35820 \, 89552 \, 23880 \, 59 \dot 7$

This sequence is A021071 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division: 

   0.01492537313432835820895522388059701...
  -----------------------------------------
67)1.00000000000000000000000000000000000...
     67
   ----
     330   210   560    640   590
     268   201   536    603   536
     ---   ---   ---    ---   ---
      620    90   240    370   540
      603    67   210    335   536
      ---    --   ---    ---   ---
       170   230   390    350    400
       134   201   335    335    335
       ---   ---   ---    ---    ---
        360   290   550    150    650
        335   268   536    134    603
        ---   ---   ---    ---    ---
         250   220   140    160    470
         201   201   134    134    469
         ---   ---   ---    ---    ---
          490   190    600   260     100
          469   134    536   201      67
          ---   ---    ---   ---     ---
           210   560    640   590    ...

$\blacksquare$

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