Reciprocal of 7

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Theorem

The decimal expansion of the reciprocal of $7$ has the maximum period, that is: $6$:

$\dfrac 1 {7} = 0 \cdotp \dot 14285 \, \dot 7$

This sequence is A020806 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division:

   0.142857...
  ------------
 7)1.000000000
     7
     --
     30
     28
     --
      20
      14
      --
       60
       56
       --
        40
        35
        --
         50
         49
         --
          10
           7
          --
          ..

$\blacksquare$