Please prove that $(p!+1)$ is a prime, where $p$ is a prime. I got Euler's proof for the Question:
Prove that there is no largest prime.
But one of the solutions on internet gives this proof by Archimedes:

Please help me to understand the proof.

Suppose $p$ is the largest prime.
By the fundamental theorem of arithmetic, you can write $p! + 1$ as a product of primes, which are necessarily not larger than $p$, and thus are factors of $p!$.
But $p! + 1$ divided by any integer $q \le p$ gives remainder $1$, as
$$
p! + 1 = q \cdot (p \cdot \ldots \cdot (q+1) (q-1) \cdot \ldots 2 \cdot 1) + 1,
$$ and so $p! + 1$ is not divisible by $q$.
