Whats the smallest possible value off p and q?

Question

The question is taken from a problem-solving test:

Both p and q are positive integers. What's the smallest possible value of the numbers such that:

$\frac{p}{q}=0,126126\overline{126}$

I don't really know how to proceed with such a question without just trying different values I see if I find some pattern. Does anyone have a method for solving these types of questions in general?

We may have a language issue here. Can you confirm that the right hand value of the equation is a repeating decimal less than 0 with the digits 126 repeating? — S. A. Lloyd, Nov 11 '22 at 19:53
If $\frac{p}{q} = 0.126126\dots$, what can you say about $1000p$? — Patrick Stevens, Nov 11 '22 at 19:53
Using the (inftinite) geometric series: $0.\overline{126}=126\cdot \sum\limits_{k=1}^\infty \left(\frac{1}{10^3}\right)^k=126\cdot \frac{\frac{1}{10^3}}{1-\frac{1}{10^3}}=126\cdot \frac{1}{10^3-1}$. Next you cancel some numbers. — callculus42, Nov 11 '22 at 20:55

score 1 · Answer 1 · answered Nov 12 '22 at 05:14

$$\frac{p}{q}=0.\overline{126}$$ Multiply the equation above by $10^n$ where $n=$the number of repeating digits. $$1000\frac{p}{q}=126.\overline{126}$$ Subtracting the given equation from the second equation. $$999\frac{p}{q}=126$$ Solve the equation above for $\frac{p}{q}$. $$\frac{p}{q}=\frac{126}{999}$$ $$\frac{p}{q}=\frac{14}{111}$$ Therefore, $p=14$ and $q=111$ are the smallest possible integer values for $p$ and $q$.

Whats the smallest possible value off p and q?

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