Example 1: "Calculate the number of elements of order 2 in the group $C_{20} \times C_{30}$"
To do this, I split the groups into their primary decompositions and got that the groups with elements of order 2 are $C_4$ and $C_2$. From here, to then work out the number of elements of order 2 I did:
$\varphi(4) = 2$, $\varphi(2) = 1$
So number of elements of order 2 will be $(2 + 1)^2 - 1 = 3$, which was the correct answer.
However
Example 2: "Compute the number of elements of order 35 of the group $\mathrm{Aut}(C_{6125})$"
To do this, I can just check that 35 divides 6125 and then use the Euler totient function. Why do I not have to split 6125 into its primary decomposition and then use that little formula to work out the number of elements? Is it because this is a cyclic group and so I can just use the Euler function, however as the other one is a direct product, I need to use a different method?