Let $a=(1 3 5)(1 5 6)(1 3 5)$
I had to write this as a product of disjoint cycles and got $(1 5)(3 6)$ which I believe is correct.
Then figure out $a^{24}$ and $a^{25}$.
Now $a^{24}$ is the identity but would $a^{25}$ just be $(5 1)(6 3)$.
Please let me know.
In short, you are correct.
$(15)(36)$ is the right representation of $a$ as a product of disjoint cycles, and $a^{24}$ is the identity while $a^{25}$ is just $a$ again, because $a$ has order $2$. Note that it doesn't matter if we write $(51)(63)$ or $(15)(36)$ - in general cyclic permutations represent the same cycle.