List all the elements of themultiplicative group of 15

The multiplicative group of integers modulo \( n \) (denoted as \( \mathbb{Z}_n^* \)) consists of all integers from 1 to \( n-1 \) that are coprime (i.e., have a greatest common divisor of 1) with \( n \). 

For \( n = 15 \), we need to find all the elements \( a \) in the set \(\{1, 2, 3, \dots, 14\}\) such that \( \gcd(a, 15) = 1 \).

### Calculating \( \gcd(a, 15) \) for each \( a \):

- \( \gcd(1, 15) = 1 \)
- \( \gcd(2, 15) = 1 \)
- \( \gcd(3, 15) = 3 \)
- \( \gcd(4, 15) = 1 \)
- \( \gcd(5, 15) = 5 \)
- \( \gcd(6, 15) = 3 \)
- \( \gcd(7, 15) = 1 \)
- \( \gcd(8, 15) = 1 \)
- \( \gcd(9, 15) = 3 \)
- \( \gcd(10, 15) = 5 \)
- \( \gcd(11, 15) = 1 \)
- \( \gcd(12, 15) = 3 \)
- \( \gcd(13, 15) = 1 \)
- \( \gcd(14, 15) = 1 \)

### Elements of \( \mathbb{Z}_{15}^* \):
The integers \( a \) that satisfy \( \gcd(a, 15) = 1 \) are:

\[
\{1, 2, 4, 7, 8, 11, 13, 14\}
\]

These are the elements of the multiplicative group modulo 15.

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**Related Questions:**

1. What is the order of each element in the multiplicative group modulo 15?
2. How would you find the generators of the multiplicative group of 15?
3. What is the structure of the multiplicative group modulo 15?
4. How does the Euler's Totient Function relate to the size of the multiplicative group?
5. How would you generalize this process for any modulo \( n \)?

**Tip:** The number of elements in the multiplicative group \( \mathbb{Z}_n^* \) is given by Euler's Totient Function \( \phi(n) \).

Explore the elements of the multiplicative group modulo 15, including integers coprime to 15. Learn how to calculate Euler's Totient Function and determine the elements of \( \mathbb{Z}_{15}^* \).

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