What does the notation $q^k || n$ mean?

Question

I am currently reading the paper "PRIMES is in p" and have come across some notation that I don't quite understand in this following sentence

Consider a prime $q$ that is a factor of $n$ and let $q^k || n$. Then...

What does the notation $q^k || n$ mean here?

The full paper can be found here and the notation described above is used in the proof on page 2

Answer 1

Typically, that $q^k$ divides $n$ but no higher power of $q$ does.

Answer 2

It means "divides exactly" in the sense that $q^{k+1}$ does not divide n.

Answer 3

It means that $k$ is the higher power that divides $n$. In other words: $$q^k\|n\iff q^k|n\wedge q^{k+1}\nmid n$$