write down all the elements of multiplicative group z16

The multiplicative group of integers modulo 16, denoted as \((\mathbb{Z}_{16})^\times\), consists of all integers less than 16 that are relatively prime to 16. In other words, the elements of this group are the numbers that have a greatest common divisor (gcd) of 1 with 16.

To find the elements:

- The integers less than 16 are: \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\).
- We check the gcd of each with 16.

The gcd is 1 for the numbers:

- \( \gcd(1, 16) = 1 \)
- \( \gcd(3, 16) = 1 \)
- \( \gcd(5, 16) = 1 \)
- \( \gcd(7, 16) = 1 \)
- \( \gcd(9, 16) = 1 \)
- \( \gcd(11, 16) = 1 \)
- \( \gcd(13, 16) = 1 \)
- \( \gcd(15, 16) = 1 \)

Thus, the elements of the multiplicative group \((\mathbb{Z}_{16})^\times\) are:

\[
\{1, 3, 5, 7, 9, 11, 13, 15\}
\]

Would you like more details or have any questions? 

Here are 5 related questions you might consider:
1. What is the order of each element in the multiplicative group \((\mathbb{Z}_{16})^\times\)?
2. How do you determine the generator of a cyclic group like \((\mathbb{Z}_{16})^\times\)?
3. How does the structure of \((\mathbb{Z}_{16})^\times\) compare with other multiplicative groups like \((\mathbb{Z}_{12})^\times\)?
4. Can you construct a subgroup of \((\mathbb{Z}_{16})^\times\) and determine its order?
5. What are the properties of the Euler's totient function applied to \(n = 16\)?

**Tip:** When analyzing multiplicative groups, always check for the gcd to identify elements that are relatively prime to the modulus.

Explore the elements of the multiplicative group Z16 and understand how to identify integers modulo 16 that are relatively prime to 16. This explanation includes detailed examples and the application of the gcd formula in number theory.

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