Find the inverse of 42 modulo 43, then modulo 59

To verify if needed:
$42 \equiv-1(\bmod 43)$, so 42 is its own inverse modulo 43.

On the other hand, there is no quick trick to determine the inverse of 42 modulo 59; one must go through the extended Euclidean algorithm, and we find that the inverse of 42 modulo 59 is 52.

## Theorem 13.

Wilson For $n$ an integer we denote $n!=1 \times 2 \times 3 \times \cdots \times(n-1) \times n$. (pronounced "factorial $n$ "). If $p$ is prime, then $(p-1)!\equiv-1(\bmod p)$.