The group $\mathbb{Z}_{10}$ has precisely $4$ subgroups True/False: The group $\mathbb{Z}_{10}$ has precisely $4$ subgroups.
Solution: 
True, since there are $4$ divisors $1,2,5,10$ thus it has $4$ subgroups.

You are correct.
If you define $\mathbb{Z}_{10} = \mathbb{Z}/ 10\mathbb{Z}$, then this follows from the correspondence theorem: the subgroups of $\mathbb{Z}_{10}$ correspond to the subgroups of $\mathbb{Z}$ that contain $10\mathbb{Z}$, and there is exactly one such subgroup for each divisor of $10$.
