Reciprocal of 21

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Theorem

The decimal expansion of the reciprocal of $21$ is as follows:

$\dfrac 1 {21} = 0 \cdotp \dot 04761 \, \dot 9$

This sequence is A021025 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division:

   0.04761901...
  --------------
21)1.00000000000
     84
     --
     160
     147
     ---
      130
      126
      ---
        40
        21
       ---
        190
        189
        ---
          100
           84
          ---
          ...

$\blacksquare$


Sources