The remainder when $x^{9999}+x^{8888}+x^{7777}+\mathrm{.......}+x^{1111}+1$ is divided by $x^9+x^8+x^7+\mathrm{.......}+x^1+1$  is

$\begin{array}{l}P\left(x\right)=x^{9999}+x^{8888}+x^{7777}+\mathrm{.......}+x^{1111}+1\\ Q\left(x\right)=x^9+x^8+x^7+\mathrm{.......}+x^1+1\\ P\left(x\right)-Q\left(x\right)=(x^{9999}-x^9)+(x^{8888}-x^8)+\mathrm{......}+(x^{1111}-x^1)\\ x^9(x^{9990}-1)+x^8(x^{8880}-1)+\mathrm{......}+x(x^{1110}-1)\\ x^9(x^{10}-1)Q_1\left(x\right)+x^8(x^{10}-1)Q_2\left(x\right)+\mathrm{.......}+x^{10}(x^{10}-1)Q_0\left(x\right)\\ =(x^{10}-1)(x^9{Q}_1\left(x\right)+\mathrm{.....}+1)\\ =(x-1)s\left(x\right)g\left(x\right)\\ \therefore (x-1)\text{is a factor of}f\left(x\right)\\ P\left(x\right)-Q\left(x\right)=(x-1)s\left(x\right)g\left(x\right)\end{array}$