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.