Infinitely many primes $p=6r-1$ such that $(k,r) = 1$. Let $k$ be a positive integer. Is it true that there are infinitely many primes of the form $p=6r-1$ such that $r$ and $k$ are coprime? Feel free to assume any well known (even if hard to prove) theorem such as Dirichlet's theorem for arithmetic progressions.
Update: can we do it only with assuming the version that for any coprime $a$ and $d$ there are infinitely many primes $p$ with $p\equiv a \pmod d$?
Any help appreciated!
 A: Denote $(a,b)$ by the greatest common divisor of $a$ and $b$. Denote also $q^a||k$ when $q^a$ is the largest power of a prime  $q$ that divides $k$. 
We apply Chinese Remainder Theorem. Find a solution to the system of congruences, 
$$
x\equiv 1 \ (\textrm{mod} \ q^a) \ \textrm{for primes powers }q^a ||k, \ q\neq 5  \ \ (1),
$$
$$
x\equiv 2 \ (\textrm{mod} \ 5^b) \ \textrm{ if } 5^b ||k \ \ (2).
$$
This system is solvable, we have $x\equiv x_0$ mod $k$. Note that if $5\nmid k$ then we do not use (2) and just have $x\equiv 1$ mod $k$ with $x_0=1$. 
Then we see that $(x_0,k)=1$, so $(k,kx+x_0)=1$ and we take $r=kx+x_0$ so that $p=6(kx+x_0)-1=6kx+6x_0-1$.
By the above construction, we have $(6k, 6x_0-1)=1$. Thus, by Dirichlet's theorem, there are infinitely many $x$ such that $6kx+6x_0-1$ is prime. 
A: Assume the contrary, then there is only finitely many primes of the form $6r-1$ , where gcd$(k,r )=1$, then there is infinitely many primes of the form $6qr-1$ where the integer $q$ is a prime dividing both $k$ and $r$ , as it stands Dirichlet states : 
$$ \bigg |\{n\in \mathbb N / an+b \in \mathbb P\}\bigg | \sim n\phi(a)$$
$$ \bigg |\{n\in \mathbb N / 6n-1 \in \mathbb P\}\bigg | \sim n\phi(6)$$
but because of our assumption 
$$ \bigg |\{n\in \mathbb N / 6n-1 \in \mathbb P\}\bigg | \sim \bigg | \{n\in \mathbb N / 6qn-1 \in \mathbb P\}\bigg | \sim n\phi(6q)$$
because : $\bigg |\{n\in \mathbb N / 6n-1 \in \mathbb P\} - \{n\in \mathbb N / 6qn-1 \in \mathbb P\} \bigg |$ is finite .
therefore $$\phi(6q)=\phi(6) $$ which is absurd
Conclusion : There exists infinitely many primes such that $(k,r)=1$
