$6 .(a, b)$ represents the greatest common divisor of the natural numbers $a, b$. Let $(a, b)=1$, then $\left(a^{2}+b^{2}, a^{3}+b^{3}\right)$ is ( ).
(A) 1
(b, 2
(C) only 1 or 2
(D) possibly greater than 2
6. (C).

According to the Euclidean algorithm: If $a=b q+r$, then $(a, b)=(b; r)$. Since $a^{3}+b^{3}=\left(a^{2}+b^{2}\right)(a b)-a b(a+b)$, we have $\left(a^{3}+b^{3}, a^{2}+b^{2}\right)=\left(a^{2}+b^{2}, a b\left(a+b\right)\right)$; given $(a, b)=1$. Therefore,
$$
\begin{array}{l}
\left(a^{2}+b^{2}, a b(a+b)\right)=\left(a^{2}+b^{2}, a+b\right) . \\
\text { Also, } \because a^{2}+b^{2}=(a+b)(a-b)+2 b^{2}, \\
\therefore\left(a^{2}+b^{2}, a+b\right)=\left(a+b, 2 b^{2}\right) . \\
\text { And }\left(a+b, b^{2}\right)=1, \\
\therefore\left(a+b, 2 b^{2}\right)=(a+b, 2)=1 \text { or } 2 .
\end{array}
$$