Definition of Coprime

Question

What is the exact definition of coprime? I know that it basically means that there are no common factors, and one is not a multiple of the other. However, i recently learned that 1 is coprime to any number. What is really the definition of coprime? Which of the following values are coprime?

0, 0
0, 1
0, 2
1, 1
1, 2
2, 2

What about negatives?

Ps: This is in the context of MathCounts, a middle school math competition that uses mostly real numbers (a little bit of complex too). Hope this helps.

Thanks!

Answer 1

In number theory, two integers $a$ and $b$ are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is $1$. (Wikipedia definition)

As any integer is divisible by $1$ and $1$ itself is only divisible by itself as positive integer, this trivially makes $1$ coprime with any integer.

Now as $0$ is divisible by any positive integer, it will never be coprime with any integer not equal to $1$.

So you get: no, yes, no, yes, yes, no

This definition is due to "positive integer" also applicable to negative integers.

Answer 2

I guess the definition you are referring is:

Let $a,b\in \mathbb{N}$.
$a$ and $b$ are said to be coprime if there exists $p,q\in \mathbb{Z}$ such that $ap+bq=1$.

If so it is clear that all pairs containing $1$ is coprime, and the others are not coprime (since they are both even numbers so you cannot generate odd numbers from them).