10.31. What remainder does $x+x^{3}+x^{9}+x^{27}+x^{81}+x^{243}$ give when divided by $(x-1) ?$
10.31. Answer: 6. Let $x+x^{3}+x^{9}+x^{27}+x^{81}+x^{243}=P(x) \times$ $\times(x-1)+r$. By setting $x=1$, we get $r=6$.