Step 1: Understanding the Topic
The question asks for the number of automorphisms of the cyclic group $\mathbb{Z}_{30}$. An automorphism is an isomorphism from a group to itself. For a cyclic group $\mathbb{Z}_n$, an automorphism is determined by mapping a generator (like 1) to another generator of the group.
Step 2: Key Formula or Approach
The number of automorphisms of $\mathbb{Z}_n$, denoted $|\text{Aut}(\mathbb{Z}_n)|$, is equal to the number of generators of $\mathbb{Z}_n$. The generators of $\mathbb{Z}_n$ are the integers $k$ such that $1 \le k<n$ and $\gcd(k, n) = 1$. The number of such integers is given by Euler's totient function, $\varphi(n)$.
So, we need to calculate $\varphi(30)$.
Step 3: Detailed Calculation
A. Find the prime factorization of n=30:
\[
30 = 2 \times 3 \times 5
\]
B. Apply the formula for Euler's totient function:
The function $\varphi(n)$ is multiplicative, so $\varphi(abc) = \varphi(a)\varphi(b)\varphi(c)$ if a, b, c are pairwise coprime. For a prime $p$, $\varphi(p) = p-1$.
\[
\varphi(30) = \varphi(2) \times \varphi(3) \times \varphi(5)
\]
\[
= (2-1) \times (3-1) \times (5-1)
\]
\[
= 1 \times 2 \times 4 = 8
\]
Alternatively, using the formula $ \varphi(n) = n \prod_{p|n} (1 - 1/p) $:
\[
\varphi(30) = 30 \left(1 - \frac{1}{2}\right) \left(1 - \frac{1}{3}\right) \left(1 - \frac{1}{5}\right)
\]
\[
= 30 \times \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5}
\]
\[
= \frac{30 \times 1 \times 2 \times 4}{2 \times 3 \times 5} = \frac{240}{30} = 8
\]
Step 4: Final Answer
The number of automorphisms of the cyclic group $\mathbb{Z}_{30}$ is 8.
\[
\boxed{8}
\]