Modular Arithmetic/Wilson's Theorem

Examples
5 is a prime number because,
 * $$4!=24$$

and
 * $$24\equiv-1\pmod5\implies25|5$$

which is true. 6, on the other hand, is not, as
 * $$5!=120$$

and
 * $$120\equiv-1\pmod 6\implies 121|6$$

which is false.