## 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
$1 < k < n$
.