10. Let $p$, $q$, $r$ be prime numbers, and
$$
p|(q r-1), q|(r p-1), r \mid(p q-1) \text {. }
$$

Find all possible values of $p q r$.
10. From the given, we know
$$
\begin{array}{l}
p q r \mid(q r-1)(r p-1)(p q-1) . \\
\text { Then }(q r-1)(r p-1)(p q-1) \\
=p^{2} q^{2} r^{2}-p q r(p+q+r)+p q+q r+r p-1 \\
\Rightarrow p q r 1(p q+q r+r p-1) \\
\Rightarrow \frac{p q+q r+r p-1}{p q r}=\frac{1}{p}+\frac{1}{q}+\frac{1}{r}-\frac{1}{p q r} \text { is }
\end{array}
$$

a positive integer.
Let $k=\frac{1}{p}+\frac{1}{q}+\frac{1}{r}-\frac{1}{p q r}$.
Since $p, q, r \geqslant 2$, we have $1 \leqslant k \leqslant 3$. If $q \geqslant 5$, then $k \leqslant \frac{1}{2}+\frac{2}{5}1$, which is also a contradiction. Therefore, $q=3$.
From $1=\frac{1}{2}+\frac{1}{3}+\frac{1}{r}-\frac{1}{6 r}$, we get $r=5$.
Upon verification, $p=2, q=3, r=5$ satisfies the condition.
In conclusion, $p q r=30$.