For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$. Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$
Source: 2024 CWMO P1
Tags: number theory
For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$. Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$