6. If $a, b \in \mathbf{Z}^{+}$, and satisfy (1) $a b(a+b)$ is not divisible by 7; (2) $(a+b)^{7}-a^{7}-b^{7}$ is divisible by 7, then the minimum value of $a b$ is $\qquad$ .
6. 18 .
$$
\begin{array}{l}
\text { Solution: }(a+b)^{7}-a^{7}-b^{7}=7 a b\left[\left(a^{5}+b^{5}\right)+3 a b\left(a^{3}\right.\right. \\
\left.\left.+b^{3}\right)+5 a^{2} b^{2}(a+b)\right]=7 a b(a+b)\left(a^{2}+b^{2}+a b\right)^{2} \text {, } \\
\therefore 7^{6} \mid\left(a^{2}+b^{2}+a b\right)^{2} \quad \text { i.e., } 7^{3} \mid a^{2}+b^{2}+a b . \\
\therefore \text { Let } a^{2}+b^{2}+a b=7^{3} \text {, i.e., }(a+b)^{2}-a b=343 \text {. } \\
\text { Therefore, }(a+b)_{\min }=19 \text { at this time } a b=18 .
\end{array}
$$