If $p,q,r$ are all primes,and $p|qr-1$,$q|pr-1$ and $r|pq-1$,find all possible values of $pqr$. If $p,q,r$ are all primes,and
$p|qr-1$$~~~~~~~~~~~~~$$q|pr-1$$~~~~~~~~~~$ and $~r|pq-1$.
Find all possible values of $pqr$.  
My work:
$qr-1=pk_1$
$pr-1=qk_2$
$pq-1=rk_3$
From the above equations, we can conclude that either $k_1,k_2,k_3$ are all even or one of the primes is $2$.
I can also get that,
$p^2q^2r^2=(pk_1+1)(qk_2+1)(rk_3+1)$
Now, I cannot proceed. Please help!

If $p\mid qr-1$, $q\mid pr-1$ and $r\mid pq-1$, then $$pqr\mid(qr-1)(pr-1)(pq-1).$$
Rewriting the right side,
$$pqr\mid (pqr)^2-pqr(p+q+r)+pq+qr+pr-1,$$
so certainly
$$pqr\mid pq+qr+pr-1,$$
which implies $pqr\leq pq+qr+pr-1$.
Observe that the RHS will be smaller then the LHS in most cases. This gives many limitations on $p$, $q$ and $r$.
