Can this be solve using modular arithmetic? $k$ is prime $\Rightarrow$ $8k+1$ is prime Is the following statement true or false?

$\forall k  \in \mathbb{N}, k$ is prime $\Rightarrow$ $8k+1$ is prime

The answer is that the statement is false because if $k=7$, then $k$ is prime but $8k+1=57$ is not prime.
Is there are way to solve this problem using modular arithmetic? If yes, what is the process of solving it and how do I select the right modulo?

There are infinitely many primes $p\equiv1\pmod 3$ (Dirichlet's theorem). And for these primes, $8p+1\equiv 0\pmod 3$, so $8p+1$ is not prime.
I am not sure if this is what you are asking for.
