Number theory proof by counter example

Question

Give an example of two cycles of lengths $r$ and $s$ respectively whose product does not have order $lcm(r,s)$

Answer 1

How about $(1\ 2)$ and $(1\ 2)$? They have lengths $2$ and $2$ but their product has order $1 \neq \operatorname{lcm}(2, 2) = 2$.

Answer 2

Consider $(r\;r+1\;r+2\;\ldots\;r+s-1\;1)(1\;2\;\ldots\;r)=(1\;2\;\ldots\;r+s-1)$, which has order $r+s-1$. Note that this equals the lcm only if $r=1$ or $s=1$, in which case no counterexample is possible.