Express $2.\overline{137}$ in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
(N/A) Let $x = 2.137137137...$ (Equation $1$).
Since there are $3$ repeating digits after the decimal point, multiply both sides by $1000$:
$1000x = 2137.137137137...$ (Equation $2$).
Subtract Equation $1$ from Equation $2$:
$1000x - x = 2137.137137... - 2.137137...$
$999x = 2135$.
Therefore, $x = \frac{2135}{999}$.
Since there are $3$ repeating digits after the decimal point, multiply both sides by $1000$:
$1000x = 2137.137137137...$ (Equation $2$).
Subtract Equation $1$ from Equation $2$:
$1000x - x = 2137.137137... - 2.137137...$
$999x = 2135$.
Therefore, $x = \frac{2135}{999}$.
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