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Given that a number $n \equiv\{1,3,7,9\} \pmod{10} $ show that there is a multiple of $n$, $q$ that is a string of consectutive $1$s.

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Hint : A string containing of $k$ $1's$ has the form $\frac{10^k-1}{9}$ , so you have to solve the equation $10^k\equiv 1\ (\ mod\ n\ )$

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As a first approximation to an answer: If $\frac1n=0.\overline{a_1\ldots a_d}$, then $n\times(a_1\ldots a_d)=?$

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  • $\begingroup$ But how do you know that is true? I wanted to say that but I don't know why that is true. $\endgroup$ – jbbrsh1494 Nov 22 '14 at 22:03

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