$$k = \lfloor \frac{p}{i} \rfloor, j = p \bmod i$$

$$p = ki + j$$

$$ki + j \equiv 0 \pmod p$$

$$kj ^ {-1} + i ^ {-1} \equiv 0 \pmod p$$

$$i ^ {-1} \equiv -kj ^ {-1} \pmod p$$

$$i ^ {-1} \equiv -\lfloor \frac{p}{i} \rfloor (p \bmod i) ^ {-1} \pmod p$$

$$i ^ {-1} \equiv p - \lfloor \frac{p}{i} \rfloor (p \bmod i) ^ {-1} \pmod p$$