Rational numbers via equivalent classes

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Sometimes the rational numbers $\mathbb{Q}$ are defined via equivalent classes $[(a.b)]\subset\mathbb{Z}\times\mathbb{Z}$ of integers. In general we have $(a_1,b_1)\sim (a_2,b_2):\Leftrightarrow a_1b_2=a_2b_1$.

How does such a class $[(a,b)]$ looks like if $\gcd(a,b)=1$?

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The class looks like this:

$$\{\dots(-2a, -2b), (-a, -b), (a, b), (2a, 2b), (3a, 3b)\dots\}$$


Also, every equivalence class $C$ can be written as $[(a,b)]$ such that $\gcd(a,b)=1$. This is pretty easy to show and good practice!