I am self studying number theory from Tom M Apostol and I couldn't think about how to solve this particular problem.
let h and k are given positive integers (h, k) =1 and A(h, k) = { h+kx : x=0,1,2,... } .
Without assuming dirichlet theorem prove that for every integer $n\geq 1$ , A(h, k) contains infinitely many numbers relatively prime to n.
Attempt: I assumed that let there exist only finitely many terms( say m)in A(h, k) relatively prime to n in hope of finding contradiction.
So, for other integers in A(h, k) which are infinitely many say $l_i$(n, $l_i$) =d, d>1 . Intutively, one can say that n would be very large only so that's not true for every n . But I am not able to prove it rigorously.
Can you please prove it rigorously or give an alternative approach which is straightforward.
Thanks!!
