I'm having trouble in solving this problem; here is my attempt:
If $\sigma\in S_{n}$ ($S_{n}$ is the symmetric group) has order $200$, then, there exists $\sigma_{1},...,\sigma_{r}$ disjoint cycles with respective lengths of $l_{1},...,l_{r}\in\mathbb{N}$ such that $\sigma=\sigma_{1}...\sigma_{r}$ and $lcm(l_{1}...l_{r})=200$. As $200=2^{3}\cdot 5^{2}$, there must exist $i,j\in\{1,...,r\}$ (equal or different) such that $l_{i}$ is a multiple of $2^{3}$ and $l_{j}$ is a multiple of $5^{2}$. So, my aim will be to find the less $n$ where I can find that this last condition stands...and... after this disconnected ideas, I am quite lost.
I would appreciate so much some guidance on this question... Thank you so much in advance!