$(a,bc)=\frac{(a,b)(a,c)}{(a,b,c)}$ : when is this gcd-identity true?

Question

Let $a$, $b$ and $c$ be integers and let $(.,.)$ denotes the $\operatorname{gcd}$ function. When is this indentity true : $$(a,bc)=\frac{(a,b)(a,c)}{(a,b,c)} \quad ?$$

Many thanks !

Answer 1

It's true if $\,a\,$ is squarefree, else it may fail: if $\,p^2\mid a\,$ for a prime $\,p,\,$ say $\,a=\bar a p^2,\,$ and $\,b=p=c\,$

$$ p^3\mid (a,bc)(a,b,c)=(\bar ap^2,p^2)(\bar ap^2,p)\ \ \ {\rm but}\ \ \ p^3\nmid (a,b)(a,c) = (\bar ap^2,p)^2\qquad\qquad $$

However, if $\,a\,$ is squarefree then one easily checks that the identity is true for $\,a=p\,$ prime, hence it will remain true for $\,a\,$ being a product of distinct primes.