Determine the order of the element 7 in the multiplicative group of 16

To determine the order of the element \(7\) in the multiplicative group \((\mathbb{Z}_{16})^\times\), we need to find the smallest positive integer \(k\) such that:

\[
7^k \equiv 1 \ (\text{mod} \ 16)
\]

Let's compute the powers of \(7\) modulo 16:

\[
7^1 = 7 \equiv 7 \ (\text{mod} \ 16)
\]

\[
7^2 = 7 \times 7 = 49 \equiv 49 \ (\text{mod} \ 16) \equiv 49 - 3 \times 16 = 49 - 48 = 1
\]

So,

\[
7^2 \equiv 1 \ (\text{mod} \ 16)
\]

The smallest \(k\) for which \(7^k \equiv 1 \ (\text{mod} \ 16)\) is \(k = 2\).

Thus, the order of the element \(7\) in the multiplicative group \((\mathbb{Z}_{16})^\times\) is \(2\).

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. What is the order of other elements like 3 or 5 in \((\mathbb{Z}_{16})^\times\)?
2. How does the order of an element relate to the structure of the group?
3. What are the possible orders of elements in \((\mathbb{Z}_{16})^\times\)?
4. How can the concept of order be applied to find generators in cyclic groups?
5. How does the order of an element in \((\mathbb{Z}_{16})^\times\) compare with its order in other modular groups?

**Tip:** The order of an element in a group is the smallest number of times you must apply the group operation to return to the identity element.

Learn how to determine the order of the element 7 in the multiplicative group Z_16 using modular arithmetic and Euler's theorem. Understand the concepts of order in groups and solve similar problems. Suitable for advanced high school students and anyone interested in number theory and group theory.

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