Repeating Decimals With Primes

Number Theory Level 3

A repeating decimal is a rational number that has periodic decimal expansion with some period $k$ . For example, $\frac{1679}{2220}=0.756 \textcolor{#20A900}{306} \textcolor{#D61F06}{306} \textcolor{#20A900}{306} ...$ is a repeating decimal with period $3$ because the repeating part $306$ has length three.

$p$ is a prime number such that the decimal representation of the fraction $\frac{1}{p}$ is a repeating decimal with period $5$ . Find the sum of all possible values for $p$ .

The answer is 312.

1 solution

X X
Oct 28, 2018

If $n$ is a positive integer smaller than $99999$ , then $\frac1p=0.\overline{00001}\times n=\frac n{99999}$ .

So $p$ must be a divisor of $99999=3^2\times41\times271$ .

Possible $p$ s are $41,271$ ( $3$ has a period of 1), so $41+271=312$

Repeating Decimals With Primes

The answer is 312.

1 solution

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