Period of Reciprocal of 17 is of Maximal Length

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Theorem

The decimal expansion of the reciprocal of $17$ has the maximum period, that is: $16$:

$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$

This sequence is A007450 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division:


   0.058823529411764705...
  ------------------------
17)1.000000000000000000000
     85      68
     --      --    
     150      20        
     136      17
     ---      --
      140      30
      136      17
      ---      --
        40     130
        34     119
        --     ---
         60     110
         51     102
         --     ---
          90      80
          85      68
          --      --
           50     120
           34     119
           --     ---
           160      100
           153       85
           ---      ---
             70     ...
             68

$\blacksquare$


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