G2.2 If $R$ is the remainder of $1^{6}+2^{6}+3^{6}+4^{6}+5^{6}+6^{6}$ divided by 7 , find the value of $R$.
$\begin{array}{l}x^{6}+y^{6}=(x+y)\left(x^{5}-x^{4} y+x^{3} y^{2}-x^{2} y^{3}+x y^{4}-y^{5}\right)+2 y^{6} \\ 6^{6}+1^{6}=7 Q_{1}+2 ; 5^{6}+2^{6}=7 Q_{2}+2 \times 2^{6} ; 4^{6}+3^{6}=7 Q_{3}+2 \times 3^{6} \\ 2+2 \times 2^{6}+2 \times 3^{6}=2(1+64+729)=1588=7 \times 226+6 ; R=6 \\ \text { Method } 21^{6}+2^{6}+3^{6}+4^{6}+5^{6}+6^{6} \equiv 1^{6}+2^{6}+3^{6}+(-3)^{6}+(-2)^{6}+(-1)^{6} \bmod 7 \\ \equiv 2\left(1^{6}+2^{6}+3^{6}\right) \equiv 2(1+64+729) \bmod 7 \\ \equiv 2(1+1+1) \bmod 7 \equiv 6 \bmod 7 \\\end{array}$