12. (NET 1) ${ }^{\mathrm{IMO} 2}$ Find two positive integers $a, b$ such that none of the numbers $a, b, a+b$ is divisible by 7 and $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$.
12. By the binomial formula we have  
\[ \begin{aligned} (a+b)^{7}-a^{7}-b^{7} & =7 a b\left[\left(a^{5}+b^{5}\right)+3 a b\left(a^{3}+b^{3}\right)+5 a^{2} b^{2}(a+b)\right] \\ & =7 a b(a+b)\left(a^{2}+a b+b^{2}\right)^{2} \end{aligned} \]  
Thus it will be enough to find \(a\) and \(b\) such that \(7 \nmid a, b\) and \(7^{3} \mid a^{2}+a b+b^{2}\). Such numbers must satisfy \((a+b)^{2}>a^{2}+a b+b^{2} \geq 7^{3}=343\), implying \(a+b \geq 19\). Trying \(a=1\) we easily find the example \((a, b)=(1,18)\).