Question 5
Let $n$ and $a$ be positive integers such that
* $a^{2^n} - a$ is divisible by $n$,
* $\sum_{k=1}^{n} k^{2024} a^{2^k}$ is not divisible by $n$.
Prove that $n$ has a prime factor *smaller* than $2024$.
* $a^{2^n} - a$ is divisible by $n$,
* $\sum_{k=1}^{n} k^{2024} a^{2^k}$ is not divisible by $n$.
Prove that $n$ has a prime factor *smaller* than $2024$.