1. Let $f(x)=x^{99}+x^{98}+\cdots+x+1$, find the remainder when $f\left(x^{100}\right)$ is divided by $f(x)$.
$$
\text { 1. } \begin{aligned}
f\left(x^{100}\right) & =\left(x^{100}\right)^{99}+\left(x^{100}\right)^{98}+\cdots+x^{100}+1 \\
& =\left[\left(x^{100}\right)^{99}-1\right]+\left[\left(x^{100}\right)^{98}-1\right]+\cdots+\left(x^{100}-1\right)+100 \\
& =\left(x^{100}-1\right) F(x)+100 \\
& =f(x)[(x-1) F(x)]+100,
\end{aligned}
$$

Here $F(x)$ is a polynomial with integer coefficients, so the remainder when $f\left(x^{100}\right)$ is divided by $f(x)$ is 100.