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Counting messed up number theory messed up P6

Source: STEMS Math 2021-22 Cat A p6

December 20, 2021
number theory

Problem Statement

Let n=(q+2)q2021n=(q + 2)q^{2021} where q=109+7q=10^9+7. For every k<=nk<=n and prime p∣np|n, define fp,k(n)f_{p,k}(n) =vp v_{p}((nk)) (\binom{n}{k}) (vpv_{p}(i)(i) is the highest power of pp that divides ii). Let mm be the maximum possible (over all kk) value of the expression ∏p,prime,p∣nfp,k\prod_{p\text{,prime,} p|n} f_{p,k}. Find the sum of the digits of mm.
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