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Find a Multiple of Number that Only Contains Digit 1

 Any natural number coprimes with 10 has a mulitple that only contains digit 1 in its decimal representation.




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Arg(x/y) = Arg(x) - Arg(y)

  It is easy to see that \( \mathrm{arg} \left ( \frac{a}{b} \right ) = \mathrm{arg} \left ( a \right )  - \mathrm{arg} \left ( b \right )  \) for any complex number \( a \) and \( b \). But, this is not the case for the principle value that is \( \mathrm{Arg} \) does not always equal to \( \mathrm{Arg} \left ( a \right ) - \mathrm{Arg} \left ( b \right )  \).