Show that $0.\overline{076923} = \frac{1}{13}$.

(N/A) Let $x = 0.\overline{076923}$.
This can be written as $x = 0.076923076923...$ (Equation $1$).
Since there are $6$ repeating digits, multiply both sides by $10^6 = 1,000,000$:
$1,000,000x = 76923.076923076923...$ (Equation $2$).
Subtract Equation $1$ from Equation $2$:
$1,000,000x - x = 76923.076923... - 0.076923...$
$999,999x = 76923$.
$x = \frac{76923}{999999}$.
Dividing both numerator and denominator by $76923$:
$x = \frac{76923 \div 76923}{999999 \div 76923} = \frac{1}{13}$.
Thus, $0.\overline{076923} = \frac{1}{13}$.