How many elements are there in the group of units Z*m  for m =187= 11· 17. ​

To find the number of elements in the group of units \( \mathbb{Z}_m^* \) where \( m = 187 \) and \( 187 = 11 \times 17 \), we use Euler's totient function \( \phi(m) \). The totient function \( \phi(m) \) gives the number of integers from 1 to \( m-1 \) that are coprime with \( m \).

For a product of two distinct primes \( p \) and \( q \), \( \phi(m) \) is calculated as follows:

\[
\phi(m) = \phi(p \times q) = \phi(p) \times \phi(q)
\]

For a prime number \( p \), \( \phi(p) = p - 1 \). Therefore,

\[
\phi(11) = 11 - 1 = 10
\]
\[
\phi(17) = 17 - 1 = 16
\]

Now, multiply these values:

\[
\phi(187) = \phi(11) \times \phi(17) = 10 \times 16 = 160
\]

So, there are **160 elements** in the group of units \( \mathbb{Z}_{187}^* \).

Would you like more details, or do you have any questions?

Here are some related questions you might find interesting:

1. How do you compute the group of units \( \mathbb{Z}_m^* \) for \( m = p^2 \)?
2. What are the properties of the Euler's totient function \( \phi(m) \)?
3. How do you find generators of the cyclic group \( \mathbb{Z}_m^* \)?
4. What is the structure of \( \mathbb{Z}_{pq}^* \) where \( p \) and \( q \) are primes?
5. How does the Chinese Remainder Theorem relate to \( \mathbb{Z}_m^* \)?

**Tip:** The Euler's totient function \( \phi(m) \) is particularly useful when working with modular arithmetic and cryptography, especially in systems like RSA encryption.

Learn how to calculate the number of elements in the group of units \( \mathbb{Z}_{187}^* \) using Euler's totient function. Detailed explanation with step-by-step calculations for \( m = 187 = 11 \times 17 \).

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