19. Find a factor of 100140001 which lies between 8000 and 9000 .
(2 marks)
求 100140001 一個介乎 8000 和 9000 之間的因數。
（２分）
19. 8221
19. As 100140001 is of the form $10^{8}+14 \times 10^{4}+1$, we naturally try to factorise $x^{8}+14 x^{4}+1$. Indeed, we have
$$
\begin{aligned}
x^{8}+14 x^{4}+1 & =x^{8}+2 x^{4}+1+12 x^{4} \\
& =\left(x^{4}+1\right)^{2}+12 x^{4} \\
& =\left[\left(x^{4}+1\right)^{2}+4 x^{2}\left(x^{4}+1\right)+4 x^{4}\right]+8 x^{4}-4 x^{2}\left(x^{4}+1\right) . \\
& =\left(x^{4}+2 x^{2}+1\right)^{2}-4 x^{2}\left(x^{4}-2 x^{2}+1\right) \\
& =\left(x^{4}+2 x^{2}+1\right)^{2}-\left(2 x^{3}-2 x\right)^{2} \\
& =\left(x^{4}+2 x^{3}+2 x^{2}-2 x+1\right)\left(x^{4}-2 x^{3}+2 x^{2}+2 x+1\right)
\end{aligned}
$$

By putting $x=10$, we get $100140001=12181 \times 8221$. Hence the answer is 8221 .