Can Arithmetic Progressions Form Infinite Relatively Prime Subsequences?

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a question came up

"show that the arithmetic progression ax+b contains an infinite subsequence (not necessarily a progression), every two of whose elements are relatively prime."

i have a hunch that the chinese remainder theorem has something to do with this, but I'm not sure how. any thoughts?
 
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Is that true? What if a=2, b=o?
 
sorry, assuming a, b are non zero
 
Then a=2, b=2 is a counterexample. I think you really need that a and b are coprime, in which case the sequence actually contains infinitely many primes.
 
right again. its actually a two part question so it says on the top that (a,b) = 1, i forget to mention; if so (now that we finally got the problem) how is the CRT applicable here?
 
and deriving some sort of solution that does not employ dirichlet's theorem, i think, because then that would be obvious; i really do not know how the CRT can be used here.
 

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