The problem asks to "Find the number of generators and all the generators of a cyclic group $\langle a \rangle$ of order 15."

Solution:

For a cyclic group of order $n$, the number of generators is given by the Euler's totient function $\phi(n)$. The generators of the group are those elements that are coprime with $n$ (i.e., their greatest common divisor with $n$ is 1).

Step 1: Apply Euler's Totient Function $\phi(n)$

We know that $n = 15$, and we can calculate $\phi(15)$ using the prime factorization of 15: $15 = 3 \times 5$ Euler's totient function for $n$ is given by: $\phi(n) = n \left( 1 - \frac{1}{p_1} \right)\left( 1 - \frac{1}{p_2} \right)\cdots \left( 1 - \frac{1}{p_k} \right)$ where $p_1, p_2, \dots, p_k$ are the distinct prime factors of $n$.

Thus: $\phi(15) = 15 \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{5} \right) = 15 \times \frac{2}{3} \times \frac{4}{5} = 15 \times \frac{8}{15} = 8$ So, the number of generators is 8.

Step 2: Identify the Generators

The generators of a cyclic group are the elements that are coprime with 15. The elements of the group $\mathbb{Z}_{15} = \{0, 1, 2, \dots, 14\}$.

We find which of these