Question 73
Suppose $P(x)$ is a monic polynomial of degree $2023$ such that
$$
P(k) = k^{2023} P\left(1 - \frac{1}{k}\right)
$$
for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$, where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n \lt 2027$ such that $b n - a$ is divisible by the prime $2027$.
$$
P(k) = k^{2023} P\left(1 - \frac{1}{k}\right)
$$
for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$, where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n \lt 2027$ such that $b n - a$ is divisible by the prime $2027$.