The product of the $p−1$ integers between any two non-consecutive multiples of $p$ What is the meaning of this statement by Wilson'theorem:
The product of the $p−1$ integers between any two consecutive multiples
of $p$ is congruent to $−1$ modulo $p$, where $p$ is a prime.
That's:
$$(n+1)(n+2)(n+3)(n+4)\equiv-1\pmod 5$$
When we see that by example, then we write: for $n=10$, $p=5$
$$(10+1)(10+2)(10+3)(10+4)\equiv-1\pmod 5$$
My question is: where is the two consecutive multiples of $5$? Can anyone help me and give me an example of this statement: The product of the $p−1$ integers between any two non-consecutive multiples of $p$, where $p=5$?
 A: The multiples of $p$ are:
$$\{\ldots, -2p, -p, 0, p, 2p, 3p, \ldots, np, (n+1)p, \ldots\}$$ and if we pick any two consecutive multiples of $p$, they would have the form
$$np, (n+1)p$$ for some integer $n$.  Between any two such consecutive multiples are $p-1$ non-multiples
$$np + 1, np + 2, \ldots, np + (p-1).$$  Notice the last term is equal to $(n+1)p-1$.  The product of these is what is being considered in Wilson's theorem.
The reason why this interval must be chosen is because, if a set of $p-1$ consecutive integers is chosen in such a way that one of those integers is a multiple of $p$, then their product is a multiple of $p$, hence is $0$ modulo $p$, not $-1$.
A: Consecutive multiples of $5$ look like $5n$ and $5(n+1)$, so $10$ and $15$, as Slugger stated in the comments.
There are thus $5(n+1)-5n-1 = 5-1 = 4$ numbers between them, namely
$$5n+1, 5n+2, 5n+3, 5n+4$$.
Your statement about non consecutive multiples doesn't really make sense, as there would be more than $p-1=4$ numbers between them. In addition, it would include a multiple of $5$ between them so the product would be $0\bmod 5$
A: There are not p-1 integers between two non-consecutive multiples if p, but at least 2p-1. For example 9 integers between 110 and 120 which are non-consecutive multiples of 5.
Since there is at least one multiple of p between two non-consecutive multiples of p (that’s what makes them non consecutive, the fact that there is another one in between), the product is a multiple of p, therefore 0 modulo p.
