ATRACTOR

Let us now see what happens if \(a=b+c\).

If \(d\) is a common divisor of \(b\) and \(c\), there exist \(k\) such that \(b=kd\) and \(l\) such that \(c=ld\). Then: \[a=kd+ld=(k+l)d,\] so, \(d\) is a divisor of \(a\) (and, hence, it is a common divisor of \(a\) and \(b\), since \(d\) was already a divisor of \(b\)).

Hence, the common divisors of \(b\) and \(c\) are all common divisors of \(a\) and \(b\).

What happens if \(d\) is a common divisor of \(a\) and \(b\)? If \(a=kd\) and \(b=ld\), then \(kd=ld+c\) (because \(a=b+c\)), hence \(c=(k-l)d\), which means that \(d\) is a divisor of \(c\) (and so a common divisor of \(b\) and \(c\)).

Conclusion: the common divisors of \(a\) and \(b\) are exactly the common divisors of \(b\) and \(c\) (and so, in particular the greatest common divisor of \(a\) and \(b\) is the greatest common divisor of \(b\) and \(c\)).