What operations is this set closed under? The set of all $\{(a,b) : (a,b) \in \Bbb{Z}^2, \gcd(a,b) \neq 1\}$.

Let $S = \{(a,b): a,b \in \Bbb{Z}, \gcd(a,b) \neq 1 \}$.

Under what binary operations is $S$ closed (that we can come up with)?

I came up with the following:

\begin{align*} (a,b)(c,d) = \\ &(1) \ \ (ac, bd) \\ &(2) \ \ (ad, bc) \\ &\dots etc. \\ &(3) \ \ (ac - bd, ad + bc) \\ &(4) \ \ (lcm(ac), lcm(bd)) \\ &(5) \ \ (gcd(ac - bd, ad +bc), gcd(ac+bd, ad-bc)) \end{align*}

$(5)$ I'm not sure of, but you get the idea. What other expressible binary operations is $S$ closed under?

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