How do you show gcd( p+qi, p-qi) | 2 in this case?

Question

Consider the ring of numbers $\Bbb{Z}$ [i] = {a + bi | a, b $\in$ $\Bbb{Z}$, $i^{2}$ = 1}

Prove that if p and q are coprime integers, then gcd(p + qi, p - qi) | 2.

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so gcd(p, q) = 1, you can say there exists a,b $\in$ $\Bbb{Z}$[i] where ap + bq = 1

Any help would be appreciated thanks

Answer 1

We can see that $\,\gcd(p+qi,p-qi)$ divides both $(p+qi)+(p-qi)=2p$ and $(p+qi)-(p-qi)=2qi$. Since $\gcd(2p,2qi)=2$, we conclude that $\gcd(p+qi,p-qi) \mid 2$.