Patterns in Decmal Expansions of Fractions

There is a patter in the expression of fractions of decimals that arises because of the number of remainders that are possible.
\[\frac{1}{7}=0.142857142857...\]

The expansion repeats after six duigits. There are six possible remainders on dividing by seven. Six is a factor of six.
\[\frac{1}{9}=0.1111111...\]

The expansion repeats after one duigit. There are nine possible remainders on dividing by nine. One is a factor of nine.
\[\frac{1}{11}=0.090909090909090...\]

The expansion repeats after 2two duigits. There are ten possible remainders on dividing by eleven. Two is a factor of ten.
\[\frac{1}{13}=0.076923076923...\]

The expansion repeats after 2six duigits. There are twelve possible remainders on dividing by thirteen. Six is a factor of twelve.
In general for any odd number  
\[n\]
, if  
\[\frac{1}{n}\]
  has an infinite decimal expansion that eventually repeats after  
\[m\]
  digits, then  
\[m\]
  is a factor of  
\[m-1\]
.
The same is true for  
\[\frac{k}{n}\]
  for  
\[1 < k < n\]
.

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