Recurring Decimals From Fractions
\[\frac{5}{3}=1.6666666...\]
\[\frac{4}{7}=0.57142857142857...\]
\[\frac{7}{8}=0.875\]
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. 
