UA-UMO 2021 Problem 102

Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$ . Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$ , number $P(k)$ is divisible by the number $d$ .

(Oleksii Masalitin)