What is the equal sign with 3 lines mean in Wilson's theorem?

Question

I'm reading up on Wilson's Theorem, and see a symbol I don't know... what does an equal sign with three lines mean?

I'm looking at the example table and I still can't infer what they are trying to say about that relationship between equations.

Answer 1

$$(n-1)! \equiv -1\pmod n$$ means that $(n-1)!$ and $-1$ differ by a multiple of $n$. Or, if you prefer, that $(n-1)!+1$ is a multiple of $n$.

In general, $$a\equiv b\pmod n$$ means that $a$ and $b$ differ by a multiple of $n$, or that $a-b$ is a multiple of $n$.

It's explained in detail in the Wikipedia article on "modular equivalence". The $\equiv$ symbol itself is pronounced "is equivalent to".

Answer 2

Wilson's Theorem:

$$(n-1)!\ \equiv\ -1 \pmod n.$$

It means "is congruent to" (modulo n): that is, $$(n - 1)! -(-1) = (n - 1)! + 1 \equiv 0 \pmod n$$ And that simply means that $n$ divides $(n - 1)! + 1$.

Answer 3

a ≡ b (mod n)

is equivalent to

a mod n = b :(if b less than n)

a mod n = b mod n :(else)

(I am not sure, if also b>0).