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!

Without loss of generality, assume that $ p \le q \le r.$
Multiplying the three relations gives
$$pqr | p^2q^2r^2 - p^2qr - pq^2r - pqr^2 + pq + pr + qr - 1;$$
therefore $$pqr | pq+pr+qr - 1 < 3qr$$ and thus $p = 2$.
By Hagen's comment, $q$ and $r$ are odd.
Multiplying again gives
$$ qr | (2r - 1)(2q-1) = 4qr - 2q - 2r + 1$$ and therfore
$$ qr | 2q + 2r - 1 < 4r,$$ i.e. $ q = 3$.
Finally, the last relation now reads $r | 6 - 1 = 5$ giving $r = 5$.
This means that $pqr = 30$ in all cases.
