3
$\begingroup$

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?

$\endgroup$
  • 5
    $\begingroup$ Yes and yes and yes. $\endgroup$ – WhatsUp Jul 4 at 15:31
  • 1
    $\begingroup$ 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. $\endgroup$ – lulu Jul 4 at 15:34
  • $\begingroup$ 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. $\endgroup$ – Wuestenfux Jul 4 at 15:37
  • $\begingroup$ @lulu thanks sir, your answer is what i want to ask $\endgroup$ – harezmi Jul 4 at 15:37
  • $\begingroup$ 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? $\endgroup$ – DreiCleaner Jul 4 at 15:37
2
$\begingroup$

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 $b.$

| cite | improve this answer | |
$\endgroup$
3
$\begingroup$

We may define relatively prime integers as follows.

Two integers are relatively prime if they do not have any common prime factors.

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 $3$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.