Period for decimals

Question

How many positive integers n have the decimal expansion of 1/n purely periodic with period 3. I don't completely understand what a periodic decimal is so can you help me with that too?

Answer 1

These decimal expansions have repeating strings of three digits:

$1/27=0.037037037...=0.\overline{037},$ $1/37=0.027027027...=0.\overline{027},$

$1/111=0.009009009...=0.\overline{009},$ $1/333=0.003003003...=0.\overline{003}$,

and $1/999=0.001001001...=0.\overline{001}$.

After $1000$ is divided by a factor of $999$, the remainder is $1$, so the decimal expansion repeats.

Other factors of $999,$ namely $3$ and $9$, are factors of $9$ as well,

so their decimals repeat after only $1$ digit:

$1/3=0.333...=0.\overline{3}$ and $1/9=0.111...=0.\overline{1}.$