The cyclic subgroups of \(\mathbb{Z}_{18}\) correspond to divisors of 18: \(1, 2, 3, 6, 9, 18\). Each divisor \(d\) gives a unique cyclic subgroup \(\langle \frac{18}{d} \rangle\) of order \(d\). There are 6 divisors, hence 6 cyclic subgroups.