What does the notation $\equiv 1\ (\text{mod}\ p)$ mean?

Question

I'm trying to understand the Fermat theory : $a^{p-1} \equiv 1\ (\text{mod}\ p)$

I know that $a\ (\text{mod}\ p)$ gives the remainder of division of $a$ by $p$. So what is $\equiv 1\ (\text{mod}\ p)$?

Thank you.

Answer 1

$$a\equiv b\pmod p$$ means that

$a-b$ is a multiple of $p$,

or equivalently that

the remainder of dividing $a$ by $p$ is the same as the remainder of dividing $b$ by $p$.

This notation is nearly the first topic discussed in the Wikipedia article on modular arithmetic. Have you tried reading that?