Dear finite researchers of the infinite mathematics,

I have a question for you:

Given a prime $p$ and by Dirichlet a prime $q = k\cdot p+1$ - minimal of this form -, does then the number $k = (q-1)/p$ have only prime divisors $< p$?

What does the finite research literature say for such a dumb question and what do you say? If the question is not research level, then I will update the question so that its level matches the community expectations of researchness, which is to say, arbitrary rules, inventend by self-proclaimed priests of mathematics.

Given a prime $p$ and by Dirichlet a prime $q = k\cdot p+1$ - minimal of this form -, does then the number $k = (q-1)/p$ have only prime divisors $< p$? What does the research literature say for this question?