To determine the order of the multiplicative group modulo \( 20 \), denoted as \( (\mathbb{Z}/20\mathbb{Z})^* \), we need to calculate Euler's totient function \( \phi(20) \).

1. First, find the prime factorization of 20:
   \[
   20 = 2^2 \times 5
   \]
   
2. Use the formula for Euler's totient function \( \phi(n) \):
   \[
   \phi(20) = 20 \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{5}\right)
   \]
   
3. Simplify the expression:
   \[
   \phi(20) = 20 \times \frac{1}{2} \times \frac{4}{5} = 20 \times \frac{4}{10} = 8
   \]

So, the order of the multiplicative group \( (\mathbb{Z}/20\mathbb{Z})^* \) is **8**.

Would you like more details or have any questions?

Here are some related questions you might consider:
1. What are the elements of the multiplicative group modulo 20?
2. How do you calculate \( \phi(n) \) for numbers with more complex prime factorizations?
3. How does the structure of \( (\mathbb{Z}/20\mathbb{Z})^* \) compare to that of other groups like \( (\mathbb{Z}/21\mathbb{Z})^* \)?
4. How can you determine the generators of the group \( (\mathbb{Z}/20\mathbb{Z})^* \)?
5. What is the significance of the Euler's totient function in cryptography?

**Tip:** Understanding the structure of the multiplicative group modulo \( n \) is crucial in areas like cryptography, where the group order plays a key role in algorithms like RSA.

Learn how to calculate the order of the multiplicative group modulo 20 using Euler's totient function. Understand the significance of group order in number theory and its applications in cryptography and algorithms like RSA. Suitable for advanced high school students and beyond.

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