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# What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 9?

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$$What is the smallest positive integer n such that \frac{1}{n} is a terminating decimal and n contains the digit 9?$$

Guest Aug 14, 2017
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What is the smallest positive integer $$n$$ such that $$\tfrac{1}{n}$$
is a terminating decimal and $$n$$ contains the digit $$9$$ ?

Terminating decimal if and only n (denominator) is of the form $$2^r5^s$$

The smallest positive integer $$n$$:

$$\begin{array}{|rcll|} \hline 2^{12}\cdot 5^0 &=& 4096 \\ && \frac{1}{4096} = 0.000244140625 \\ \hline \end{array}$$

The smallest positive integer n is 4096.

heureka  Aug 15, 2017
edited by heureka  Aug 15, 2017

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