https://file.aiquickdraw.com/mathbot/file/1725987039822ciu5lxn3.jpg

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

Find the number of generators and all the generators of a cyclic group \( \langle a \rangle \) of order 15.

Explore how to find the number of generators and all the generators of a cyclic group of order 15. This detailed solution covers key concepts in Group Theory and Number Theory, including the application of Euler's Totient Function.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation