Period of Reciprocal of 7 is of Maximal Length

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Theorem

$7$ is the smallest integer $n$ the decimal expansion of whose reciprocal has the maximum period $n - 1$, 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).


ReciprocalOf7Cycle.png

Proof

Performing the calculation using long division:


  0.1428571
 ----------
7)1.0000000
    7
  ---
    30
    28
    --
     20
     14
     --
      60
      56
      --
       40
       35
       --
        50
        49
        --
         10
          7
         --
         .....

The reciprocals of $1$, $2$, $4$ and $5$ do not recur:

\(\ds \frac 1 1\) \(=\) \(\ds 1\)
\(\ds \frac 1 2\) \(=\) \(\ds 0 \cdotp 5\)
\(\ds \frac 1 4\) \(=\) \(\ds 0 \cdotp 25\)
\(\ds \frac 1 5\) \(=\) \(\ds 0 \cdotp 2\)


while those of $3$ and $6$ do recur, but with the non-maximum period of $1$:

\(\ds \frac 1 3\) \(=\) \(\ds 0 \cdotp \dot 3\)
\(\ds \frac 1 6\) \(=\) \(\ds 0 \cdotp 1 \dot 6\)

$\blacksquare$


Sources