Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ always relatively prime ?

share|improve this question
add comment

3 Answers 3

up vote 7 down vote accepted

The answer is clearly no: take $p=\alpha$, $q=\beta$. Then $p\beta+q\alpha=2pq$ and $q\beta=pq$.

If however $\beta$ is co-prime to $q$, then the answer is equally clearly yes, since no divisor of $q$ divides $p\beta$ and no divisor of $\beta$ divides $q\alpha$.

share|improve this answer
add comment

No, e.g. $p=3,q=5,\alpha=3,\beta=5$ is a counterexample.

share|improve this answer
add comment

Simply repeatedly apply Euclid's lemma:

$\rm\quad\quad\quad\quad\quad 1 = (pb+qa,qb)$
$\rm\quad \iff\ 1 = (pb+qa,q) = (pb,q) = (b,q)\quad\ via\quad\ (p,q) = 1$
$\quad\ $ and $\rm\ \ 1 = (pb+qa,b) = (qa,b) = (q,b)\quad\ via\quad\ (a,b) = 1$

Hence $\rm\ \ 1 = (pb+qa,qb)\ \iff\ 1 = (b,q)$

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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