\[\frac{1}{7}=0.142857142857...\]
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 \[1 < k < n\]
.