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Irreducible fraction

Find all positive integers \alpha such that it holds that

(1)   \begin{equation*} \frac{1}{\alpha} = 0.\bar{\alpha} \end{equation*}

where \overline{ \cdot } stands for the period.

Solution

Well,

    \begin{align*} \frac{1}{\alpha} = 0.\bar{\alpha} &\Leftrightarrow \frac{1}{\alpha} = 0.\alpha \alpha \alpha \dots \\ &\Leftrightarrow \frac{1}{\alpha} = \frac{\alpha}{10} \sum_{i=0}^{\infty} \frac{1}{10^i} \\ &\Leftrightarrow \frac{1}{\alpha} = \frac{\alpha}{9}\\ &\!\!\!\!\!\!\!\overset{\alpha>0}{\Leftarrow \! =\! =\! \Rightarrow } \alpha = 3 \end{align*}

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