Prove that $18!+1$ is divisible by $19$ and $23$

Question

Prove that $18!+1$ is divisible by $19$ and $23$

For $19$ we can just use Wilson's Theorem but I wasn't able to think of how to prove for 23. One way is to just multiply $(-5)(-6)(-7)(-8)(-9)(-10)(-11)(11)(10)(9)…(2)(1)$ and then find its remainder with $23$, and then add $1$ to it. But surely there's a better way right?

Answer 1

You can still use Wilson’s theorem to get started: you know that $22!\equiv -1\pmod{23}$. Then observe that

$$22!\equiv 18!(-4)(-3)(-2)(-1)\equiv\; ...\,?\pmod{23}$$