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?
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$1/27=.\overline{037}$ and $1/37=.\overline{027}$ and $1/999=.\overline{001}$ – J. W. Tanner Oct 31 '19 at 00:36
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also $1/111=.\overline{009}$, and $1/333=0.\overline{003}$ – J. W. Tanner Oct 31 '19 at 00:42
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It may be useful to know that $0.\overline{abc} = \frac{abc}{999}$, so what you're looking for is a number that divides $999$, but does not divide $99$. So factor $999$ and make some deductions. – Brian Tung Oct 31 '19 at 00:43
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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 $1/9=0.111...=0.\overline{1}.$
J. W. Tanner
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