Reciprocal of 53

Theorem

The decimal expansion of the reciprocal of $53$ is as follows:

$\dfrac 1 {53} = 0 \cdotp \dot 01886 \, 79245 \, 28 \dot 3$

This sequence is A021057 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division: 

   0.018867924528301...
  ---------------------
53)1.000000000000000000
     53    212          
     --    ---
     470    280
     424    265
     ---    ---
      460    150
      424    106
      ---    ---
       360    440
       318    424
       ---    ---
        420    160
        371    159
        ---    ---
         490     100
         477      53
         ---     ---
          130    ...
          106
          ---
           240
           212
           ---

$\blacksquare$

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