10. Let $a, b, c (1 \leqslant a < b < c \leqslant 9)$ be integers, and $\overline{a b c} \cdot \overline{b c a} \cdot \overline{c a b} + 1$ is divisible by 9. Then the minimum value of $a + b + c$ is $\qquad$, and the maximum value is $\qquad$.
10.8, 23.

Let $a+b+c \equiv r(\bmod 9)$. Then the remainders of $\overline{a b c} 、 \overline{b c a} 、 \overline{c a b}$ when divided by 9 are all $r$. Therefore,
$$
\begin{array}{l}
\overline{a b c} \cdot \overline{b c a} \cdot \overline{c a b}+1 \equiv r^{3}+1 \equiv 0(\bmod 9), \\
r \equiv 2,5,8(\bmod 9) .
\end{array}
$$

Since $a 、 b 、 c$ are three non-zero digits, we have
$$
6 \leqslant a+b+c \leqslant 24 \text {. }
$$

Combining the above results, we get
$$
(a+b+c)_{\min }=8,(a+b+c)_{\max }=23 .
$$