Recurring Decimals From Fractions

All fractions with one whole number divided by another whole number forms a sequence which eventually terminates or repeats.



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  
  then there are  
  possible remainders at each stage of long division, and they must repeat after  
  iterations of long division, so the decimal expansion is at most  
  digits long. In fct the length of the recurring expansion mus t divide  
For example

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  
  must be of at length  
  and may be  
For example
  whch repeats every three digits.
  whch repeats every 2 digits.
\[\frac{1}{37 \times 11}=0.00245700245700...\]
  whch repeats every three times two digits.

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