Question 79
For any integer $x$, let
$$
f(x) = 100! \left(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots + \frac{x^{100}}{100!}\right).
$$
A positive integer $a$ is chosen such that $f(a) - 20$ is divisible by $101^{2}$. Compute the remainder when $f(a + 101)$ is divided by $101^{2}$.
$$
f(x) = 100! \left(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots + \frac{x^{100}}{100!}\right).
$$
A positive integer $a$ is chosen such that $f(a) - 20$ is divisible by $101^{2}$. Compute the remainder when $f(a + 101)$ is divided by $101^{2}$.