Are there an infinite number of primes which are any multiple of $n$ apart?

Question

Are there an infinite number of primes which are any multiple of $n$ apart? That is take $n\in \mathbb{N}$, then is there an infinite number of primes which are separated by $\textbf{any}$ of the numbers in the below set $$ \{n,2n,3n,4n,\ldots \}. $$

Answer 1

Prime numbers are infinite, so, by the pigeonhole principle $\forall n \exists k $ s.t. there is an infinite number of primes $p\equiv k \pmod n$. Any couple among these primes has a difference which is a multiple of $n $.

Answer 2

Yes. Dirichlet's Theorem provides the proof. Let $n$ be any positive integer. Let $a$ be some number which is relatively prime to $n$. Then by Dirichlet's Theorem there are infinitely many primes in the arithmetic progression $a+kn$, $k\in \mathbb{N}.$ Any two of these primes are separated by a multiple of $n$, therefore there are infinitely many values of $k\in \mathbb{N}$ for which there is a prime gap of size $kn$.