I am trying to argue that for distinct primes $p,q,r$ we have that
$$ \gcd (pq + qr+ pr ,pqr ) = 1 = g $$
and I am wondering whether people find the following argument convincing :
Consider the prime decompositions; for $g$ to be greater than $1$ we need at least one of $p,q,r$ to occur in the decomposition of $pq + qr+ pr$. Without loss of generality suppose it is $p$ ( not necessarily exclusively ). But then the remainder modulo $p$ must be zero. However the remainder is $qr$ which is not zero modulo $p$ as $p,q,r$ are all mutually coprime. ( using $x\mid (a+b)$ and $x\mid a$ implies $x\mid b$ ) So the result follows.