Definition of q congruent to 1 (mod p)

Question

What is the meaning of q $\equiv 1\pmod p$?

If I were to write this as a definition, how would I explain with this means?

Answer 1

The meaning is that $p$ divides $q$ with remainder $1$.

Answer 2

Maybe the simplest way to phrase it would be:

If you subtract 1 from q, you get a multiple of p.

Answer 3

It means $p$ divides $q-1$.

$$q \equiv 1 \mod p$$

$$\implies q-1 \equiv 0 \mod p$$

$$\implies p | (q-1) $$