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.