List or characterize all of the units in each of the following rings.

Z10
Z12

### Units in a Ring

In any ring \( \mathbb{Z}_n \), the **units** are the elements that have a multiplicative inverse modulo \( n \). This means for an element \( a \in \mathbb{Z}_n \), \( a \) is a unit if there exists some \( b \in \mathbb{Z}_n \) such that:

\[
a \cdot b \equiv 1 \,(\text{mod}\, n)
\]

Equivalently, \( a \) is a unit if \( \gcd(a, n) = 1 \), meaning that \( a \) and \( n \) are coprime.

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### Units in \( \mathbb{Z}_{10} \)

In \( \mathbb{Z}_{10} \), the units are elements that are coprime with 10. We need to find elements \( a \in \{0, 1, 2, \dots, 9\} \) such that \( \gcd(a, 10) = 1 \).

The divisors of 10 are \( \{1, 2, 5, 10\} \), so elements that share no common divisors with 10 (other than 1) are:

- \( 1: \gcd(1, 10) = 1 \)
- \( 3: \gcd(3, 10) = 1 \)
- \( 7: \gcd(7, 10) = 1 \)
- \( 9: \gcd(9, 10) = 1 \)

Thus, the units in \( \mathbb{Z}_{10} \) are:

\[
\{1, 3, 7, 9\}
\]

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### Units in \( \mathbb{Z}_{12} \)

In \( \mathbb{Z}_{12} \), the units are elements that are coprime with 12. We need to find elements \( a \in \{0, 1, 2, \dots, 11\} \) such that \( \gcd(a, 12) = 1 \).

The divisors of 12 are \( \{1, 2, 3, 4, 6, 12\} \), so elements that share no common divisors with 12 (other than 1) are:

- \( 1: \gcd(1, 12) = 1 \)
- \( 5: \gcd(5, 12) = 1 \)
- \( 7: \gcd(7, 12) = 1 \)
- \( 11: \gcd(11, 12) = 1 \)

Thus, the units in \( \mathbb{Z}_{12} \) are:

\[
\{1, 5, 7, 11\}
\]

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### Summary of Units

- **Units in \( \mathbb{Z}_{10} \)**: \( \{1, 3, 7, 9\} \)
- **Units in \( \mathbb{Z}_{12} \)**: \( \{1, 5, 7, 11\} \)

Would you like more details on any of the steps or explanations?

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### 5 Related Questions:
1. How do we find the inverse of a unit in \( \mathbb{Z}_n \)?
2. Why is an element in \( \mathbb{Z}_n \) a unit if its greatest common divisor with \( n \) is 1?
3. What is the structure of the group of units in \( \mathbb{Z}_n \)?
4. What are the units of \( \mathbb{Z}_8 \) and how do they compare to \( \mathbb{Z}_{10} \) and \( \mathbb{Z}_{12} \)?
5. How does Euler's totient function relate to counting the number of units in \( \mathbb{Z}_n \)?

### Tip:
When working with modular arithmetic, understanding which elements are units is key to solving equations like \( ax \equiv 1 \,(\text{mod}\, n) \).

List or characterize all of the units in each of the following rings. Z10, Z12

Explore how to find the units in the rings Z10 and Z12, with a step-by-step solution and explanation. This problem covers key concepts in Abstract Algebra, including modular arithmetic and the properties of units in a ring. Suitable for undergraduate students.

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