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.