## Recurring Decimals From Fractions

All fractions with one whole number divided by another whole number forms a sequence which eventually terminates or repeats.
$\frac{5}{3}=1.6666666...$

$\frac{4}{7}=0.57142857142857...$

$\frac{7}{8}=0.875$

If the denominator is a poer of 2 or 5, or any multiple of a power of 2 by a power of 5, then the decimal expansion terminates.
If the denominator is an odd prime number
$p$
then there are
$p-1$
possible remainders at each stage of long division, and they must repeat after
$p-1$
iterations of long division, so the decimal expansion is at most
$p-1$
digits long. In fct the length of the recurring expansion mus t divide
$p-1$
.
For example
$\frac{3}{11}=0.27272727..$

The length of the recurring expansion is 2, which divides 11-1=10.
In fact a slight extension of the same remainder argument gives that if the denominator is a product of different odd primes
$p, \: q$
then the decimal expansion of
$\frac{a}{pq}$
must be of at length
$(p-1)(q-1)$
and may be
$\frac{(p-1)(q-1)}{n}$
where
$n$
divides
$(p-1)(q-1)$
.
For example
$\frac{1}{37}=0.027027027...$
whch repeats every three digits.
$\frac{1}{11}=0.09090909090...$
whch repeats every 2 digits.
$\frac{1}{37 \times 11}=0.00245700245700...$
whch repeats every three times two digits.