Let $C$ be a conjugation class in $S_n$ represented by an even element $c_1c_2\cdots c_m$, where the $c_i$ are disjoing cycles with size $r_1,\dots, r_m$. Show that if $C_{S_n}(c)\leq A_n$ then $r_1,\ldots,r_m$ are odd and distinct.
I don't get how the fact that the permutations in the centralizer are all even should help, so I don't know where to begin.
Suppose $r_i$ is even. Then $c_i$ is an odd cycle meaning $c_i\not\in A_n$ (a cycle is even iff its length is odd) and it commutes with $c=c_1...c_m$ (because all other cycles in this cycle decomposition commute with $c_i$), so $C(c)\not\subseteq A_n$. So all $r_i$ are odd.
Now suppose that, say, $r_i=r_j=r$. WLOG we can assume $i=1,j=2$, $c_1=(1,2,...r), c_2=(r+1,...,2r)$, $r$ is odd by 1. That is because every permutation with the same cycle structure as $c$ is conjugated with $c$. Then take the permutation $d=(1,r+1)(2,r+2)...(r, 2r)$. Then $d$ is odd (it is a product of an odd number of involutions) but $d^{-1}cd=c$, so, again, $C(c)\not\subseteq A_n$.