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

$$ p = ki + j $$

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

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

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

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

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