All right, so I have a problem which keeps reoccuring and I am not sure about the conclusion I found.

Say we have two sets of numbers, for example:

a ~= 6 mod 7
b ~= 5 mod 11

Then you can conclude gcd(a, b) = 1, right? I would think this is true because the numbers 5 and 6 are relatively prime and id. with 7 and 11. This of course does not mean there are no numbers in the sets which aren´t relatively prime (like 20 and 60), but this is of no importance since we are comparing the entire sets (?).

A similar example is one with:

c ~= 4 mod 6
d ~= 6 mod 10

Here gcd(6, 4) = 2 and gcd(6, 10) = 2. So compairing these sets we should get gcd(c, d) = 2 and not a greater equal number.

So does anyone think this conclusion is sound? And if so, is there a more solid way of proving this?

Thanks in advance.