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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 ?

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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 $q\alpha$.

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

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

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