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.