What would it mean, if I wrote GCD(A, m) = 1 where A would be a set of integers and m - another integer that is not in set A. Would that all values of A are coprime to m, but A values between each other don't need to be coprime? Or would it mean that values in A are coprime to each other as well?

Question

zyberg · Answer

Or is it not a valid statement at all?

kainui · Answer

I think the first way you said is fine, I've never seen it before but it seems like a completely reasonable definition to say GCD(A,m)=1 means all the elements of the set A are relatively prime to m.

This is a fun and interesting trap to fall into, I have considered before that coprime is transitive, so that if GCD(a,b)=1 and GCD(b,c)=1 then does that mean GCD(a,c)=1? The answer is NO!

For instance, let's say your set A has the members 4 and 6. then we know GCD(A,7)=1 because GCD(4,7)=1 and GCD(6,7)=1. But GCD(4,6)=2, so we know that just cause all the numbers in A are relatively prime to m, it doesn't mean they're relatively prime to each other.

rsadhvika · Answer

Perhaps below is more clear :
$$\gcd(a,m)=1$$
where $a \in A$

rsadhvika · Answer

For a specific example,
here $A$ could be a set of all odd integers and $m=2$