Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.