Let $A(h,k) = \{h + km: m = 0,1,2,\dots\}\;\;$ (EDIT: and $(h,k)=1$)
Without using Dirichlet's Theorem,
Prove that for every positive integer $n$, $A(h,k)$ contains infinitely numbers relatively prime to $n$.
Prove that $A(h,k)$ contains an infinite subset $\{a_1, a_2,\dots\}$ such that the $a_i$'s are pair-wise relatively prime.
How do I do it without Dirichlet's theorem? I tried to assume for a contradiction, but I couldn't get one.
Thank you all for your help!
We solve (1) to show the kind of tools one can use. We suppose that $k\gt 0$.
Let $m\gt n$, and let $P_m$ be the product of all the primes $\le m$. Let $x_m=\frac{P_m}{w}$, where $w$ is the product of the primes that divide both $h$ and $n$. We show that $h+kx_m$ is relatively prime to $n$.
For suppose $h+kx_m$ and $n$ are not relatively prime. Then there is a prime $p$ that divides both $n$ and $h+kx_m$.
So $p$ divides $n$. If $p$ also divides $h$, then since $h$ and $k$ are relatively prime, $p$ does not divide $k$. By the choice of $x_m$, $p$ does not divide $x_m$. So $p$ does not divide $kx_m$, and therefore $p$ does not divide $h+kx_m$.
If $p$ does not divide $h$, then $p$ divides $x_m$, so $p$ divides $kx_m$. Then again $p$ does not divide $h+kx_m$.
Since $x_m$ can be arbitrarily large, the result follows.
