Could you please define what $0,167a_{1}a_{2}...$ is? I'm sorry, here in the US we most likely use a different notation

Oh, it's nothing but $\frac{m}{n}= 0.167a_{1}a_{2}...$. In some countries they use decimal comma, like in some (USA) they use decimal point. So we have $\frac{1000m}{n}= 167.a_{1}a_{2}...$, hence $167 \leq \frac{1000m}{n} \leq 168$, or $0 \leq 1000m - 167n \leq n$. This further writes as $0 < 2n \leq 1000(6m - n) \leq 8n$, whence $6m-n > 0$, so $6m-n \geq 1$. But then $1000 \leq 8n$, i.e. $125\leq n \leq 100$, absurd.

Thank you mavropnevma. Indeed, the notation is exactly as mavropnevma specificed it.