# Can negative integers be relatively prime?

We know that if $$\gcd(a,b)$$ is equal to $$1,$$ then they are relatively prime.

However, I have seen all pairs $$(a,b)$$ as positive integers. My question is can any pair $$(a,b)$$ of negative integers be relatively prime?

For example, are $$(-1,-1)$$ and $$(-18,-5)$$ relatively prime?

• Yes and yes and yes. Jul 4, 2020 at 15:31
• Well, it depends on exactly what you mean. I expect that any mathematician would define $\gcd(a,b)$ to be the greatest integer $d$ which divides both $a$ and $b$, in which case $a, b$ could certainly be negative. But many standard software systems require positivity. I just checked, and Wolfram seems to have no problem with negative arguments, but Excel reported an error.
– lulu
Jul 4, 2020 at 15:34
• If the gcd is equal to 1, not zero. The gcd is uniquely determined modulo units in the underlying ring, here ring of integers and the units there are +1 and -1. Jul 4, 2020 at 15:37
• @lulu thanks sir, your answer is what i want to ask Jul 4, 2020 at 15:37
• For those who are answering, what about pairs like $(+a,-a)$? Does it matter if $a=1$, or $a$ is a prime number, or $a$ is a composite number? Jul 4, 2020 at 15:37

For example $$5$$ and $$-12$$ are relatively prime because the only prime factor of $$5$$ is $$5$$ which is not a facotor of $$-12$$ whose prime factors are $$2$$ and $$3$$.
Given two integers $$a$$ and $$b,$$ we can obtain $$\gcd(a, b) = d$$ by running the Euclidean Algorithm. Explicitly, there exist integers $$q_0$$ and $$r_0$$ such that $$a = bq_0 + r_0$$ and $$0 \leq r_0 \leq |b| - 1.$$ Given that $$r_0$$ is nonzero, there exist integers $$q_1$$ and $$r_1$$ such that $$b = r_0 q_1 + r_1$$ and $$0 \leq r_1 \leq r_0 - 1.$$ Continuing in this way, there exists an integer $$k$$ such that $$r_k \geq 1$$ and $$r_{k + 1} = 0$$ by the Well-Ordering Principle. One can prove that $$r_k = \gcd(a, b).$$
Consequently, the greatest common divisor can be defined for any pair of integers $$(a, b)$$ regardless of their respective signs. We say that two integers $$(a, b)$$ are relatively prime whenever $$\gcd(a, b) = 1,$$ so again, this does not depend on the sign of $$a$$ and $$b.$$