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Considering a natural integer $n$ ($n>1$). Is there a name for the set of all rationals which can be written as $p/q$ with $p$ and $q$ integers and $q$ coprime with $n$ ?

For $n=2$ it would be all the rationals with an odd denominator, for $n=3$ all rationals whose denominator is not congruent to zero mod 3, etc.

Anne Aunyme
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  • This is probably a bit too abstract for your purposes, but there's a notion called localization where we have a structured way of adjoining denominators to a ring. If $S$ is the set of numbers sharing no prime factors with $n$, then $S^{-1}\mathbf{Z}$ is the set you are describing. If I wished to avoid all the machinery involved in this, I would probably just give such a set a name; e.g. $X_n$. –  Sep 05 '23 at 01:56
  • See here. There is no standard name (but there is standard notation using localization language). – Bill Dubuque Feb 15 '24 at 13:11

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