The multiplicative group modulo \( n \), denoted as \( (\mathbb{Z}/n\mathbb{Z})^* \), consists of the integers from 1 to \( n-1 \) that are relatively prime to \( n \), with multiplication as the group operation. The order of this group is given by Euler's totient function \( \phi(n) \), which counts the number of integers less than \( n \) that are relatively prime to \( n \).

For \( n = 21 \):

1. The prime factorization of 21 is \( 21 = 3 \times 7 \).
2. The Euler's totient function \( \phi(n) \) is calculated as:

\[
\phi(21) = 21 \left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{7}\right) = 21 \times \frac{2}{3} \times \frac{6}{7} = 12
\]

Thus, the order of the multiplicative group \( (\mathbb{Z}/21\mathbb{Z})^* \) is **12**.

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 21?
2. How does the prime factorization of a number affect the Euler's totient function?
3. What is the structure of the group \( (\mathbb{Z}/21\mathbb{Z})^* \)?
4. How do you find the order of an individual element in \( (\mathbb{Z}/21\mathbb{Z})^* \)?
5. What are some applications of Euler's totient function in number theory?

**Tip:** The order of an element in a group is the smallest positive integer \( k \) such that \( a^k \equiv 1 \pmod{n} \).

Explore the order of the multiplicative group modulo 21 using Euler's totient function. Learn about prime factorization, how it affects φ(n), and discover the structure of (Z/21Z)*. Suitable for advanced high school students and enthusiasts of number theory.

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