# Period of Reciprocal of 7 is of Maximal Length

## Theorem

$7$ is the smallest integer $n$ the decimal expansion of whose reciprocal has the maximum period $n - 1$, that is: $6$:

$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$

## 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 $1$:

 $\ds \frac 1 3$ $=$ $\ds 0 \cdotp \dot 3$ $\ds \frac 1 6$ $=$ $\ds 0 \cdotp 1 \dot 6$

$\blacksquare$