Order of the element 7 in the multiplicative group of 16

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

\[
7^k \equiv 1 \pmod{16}
\]

Let's calculate \(7^k \mod 16\) for increasing values of \(k\):

1. \(7^1 = 7 \equiv 7 \pmod{16}\)
2. \(7^2 = 49 \equiv 49 \mod 16 = 1 \pmod{16}\)

Since \(7^2 \equiv 1 \pmod{16}\), the smallest \(k\) for which \(7^k \equiv 1 \pmod{16}\) is \(k = 2\).

**Conclusion**: The order of the element \(7\) in the multiplicative group \(\mathbb{Z}_{16}^*\) is \(2\).

Would you like further details or have any questions?

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Here are some related questions:
1. What is the definition of a multiplicative group \(\mathbb{Z}_n^*\)?
2. How is the order of an element in a group related to group theory?
3. What are the orders of other elements in \(\mathbb{Z}_{16}^*\)?
4. How does the structure of \(\mathbb{Z}_{16}^*\) compare to other groups like \(\mathbb{Z}_{15}^*\)?
5. What is the significance of the order of an element in modular arithmetic?

**Tip**: In modular arithmetic, the order of an element divides the Euler's totient function \(\phi(n)\) of the modulus \(n\).

Learn how to find the order of the element 7 in the multiplicative group of integers modulo 16. Understand the concepts of modular arithmetic and group theory with detailed calculations and explanations. Suitable for advanced high school students and beyond.

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