The restricted solutions of \(ax+by=\gcd(a,b)\) (Q2518181)

Let \(D\) be an integral domain with unit element, fix \(a,b,d\in D\) with \(\gcd(a,b)=d\). It is shown that there exist \(x,y\in D\) with either \(\gcd(a,y)=1\) or \(\gcd(b,x)=1\) such that \(ax+by=d\). There exist \(x,y\in D\) with both \(\gcd(a,y)=1\) and \(\gcd(b,x)=1\) such that \(ax+by=d\), if and only if for each prime divisor \(p\) of \(d\) with complete set of residues modulo \(p\) containing exactly two elements, the power of \(p\) appearing in the factorization of \(a\) is different from that of \(b\). An application is given to double loop networks.