Competition problem — Number theory

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Number theory Difficulty 5.1 AIME, harder Prove it Singapore

A student divides an integer $m$ by a positive integer $n$ , where $n \le 100$ , and claims that
$\frac{m}{n} = 0.167a_1a_2\dots$
Show the student must be wrong.

Solution

We have
$0.167 \le \frac{m}{n} < 0.168 \Rightarrow 167n \le 1000m < 168n.$
Multiply by $6$ , we get
$1002n \le 6000m < 1008n \Rightarrow 6000m - 1000n < 8n \le 800.$
But $6000m - 1000n \ge 2n > 0$ . Thus $6000m - 1000n \ge 1000$ since it is a multiple of $1000$ . We thus get a contradiction.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.