\[\frac{5}{3}=1.6666666...\]

\[\frac{4}{7}=0.57142857142857...\]

\[\frac{7}{8}=0.875\]

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.