To find the reciprocal of the fraction in the equation \( 23(x - 4) = -10 \), we first need to identify the fraction.
The fraction here is \( \frac{1}{23} \) since the equation can be rewritten by dividing both sides by 23:
\[ x - 4 = \frac{-10}{23} \]
The reciprocal of \( \frac{1}{23} \) is \( 23 \).
Thus, the reciprocal is 23.