\[\frac{1}{7}=0.142857142857...\]

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 \[1 < k < n\]

.