Raise permutation to the power of 150

Since the two permutations, let's call them $\sigma$ and $\tau$, are disjoint and hence commute, this warrants pulling the $150$ through: $(\sigma\tau)^{150}=\sigma^{150}\tau^{150}$, which in general is not true. The order of each of $\sigma$ and $\tau$ is $5$ (the length of the cycle). Hence $\sigma$ and $\tau$ to the power of any multiple of $5$ will give you the identity. So the answer is $(1)$.