How do you do this mentally?

$\begin{aligned} \chi & = \frac {99999919}{97} \\ & = \frac {10^8-81}{100-3} \\ & = \frac {100^4-3^4}{100-3} \\ & = \frac {\left(100^2-3^2\right) \left(100^2+3^2\right)}{100-3} \\ & = \frac {\left(100-3\right)\left(100+3\right) \left(100^2+3^2\right)}{100-3} \\ & = \left(100+3\right) \left(100^2+3^2\right) \\ & = \left(100+3\right) \left(10009\right) \\ & = 1000900 + 30027 \\ & = \boxed{1030927} \end{aligned}$

We observe that $\large\frac{99999919}{97}$ is equivalent to $\dfrac{100^{4}-3^{4}}{100-3}$

$\large x^{4}-y^{4}=(x-y)(x^{3}+x^{2}y+xy^{2}+y^{3})$

Therefore, $\large\frac{99999919}{97}=100^{3}+100^{2}×3+100×3^{2}+3^{3}=1000000+30000+900+27=1030927$