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How to determine if the expansion of $1/n$ would be a recurring decimal expansion or not? for example, $1/7 = 0.\overline{142857}$ but $1/8=0.125$.

So, how to find if $1/n$ would be a recurring decimal expansion or not?

Note: Here, $1/6=0.16666\ldots$ is not a recurring decimal expansion.

Michael Hardy
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anshabhi
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1 Answers1

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It will be recurring as long as the denominator (in lowest terms) does not have a factor $2$ or $5$. There will be one non-recurring digit corresponding to the highest power of $2$ or $5$ in the denominator. So $6$ has one factor two and $1/6$ has one non-recurring digit. $8$ has three factors of two and $1/8$ has three non-recurring digits. $17$ has no factors of $2$ or $5$ and $1/17$ has no non-recurring digits.

Ross Millikan
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  • can you please explain for 1/10? It has two and five as its factors, but is only 0.100... so, only one recurring digit..?? – anshabhi Jun 30 '15 at 19:39
  • The number of non-recurring digits is the maximum of the power of $2$ and the power of $5$. So for $10=2^15^1$ the maximum power is $1$ and there is one non-recurring digit. – Ross Millikan Jun 30 '15 at 19:41