To find the elements of order 10 in the group \( \mathbb{Z}_{100} \) (the integers modulo 100 under addition), we need to understand the concept of order in this group.

### Steps to Find Elements of Order 10:
1. **Definition of Order**: The order of an element \(a\) in a group is the smallest positive integer \(n\) such that \(n \cdot a = 0\) (where \( \cdot \) refers to repeated addition in this case).
   
2. **Condition for Order in \( \mathbb{Z}_n \)**: For an element \(a\) in \( \mathbb{Z}_n \) to have order \(k\), the following must hold:
   \[
   \frac{n}{\gcd(a, n)} = k
   \]
   where \(n = 100\) and \(k = 10\).

3. **Equation for \(a\)**: 
   \[
   \frac{100}{\gcd(a, 100)} = 10
   \]
   Simplifying, we get:
   \[
   \gcd(a, 100) = \frac{100}{10} = 10
   \]
   So, we need to find integers \(a\) such that \( \gcd(a, 100) = 10 \).

4. **List of Elements with \( \gcd(a, 100) = 10 \)**:
   The divisors of 100 are \(1, 2, 4, 5, 10, 20, 25, 50, 100\). Now we find which elements of \( \mathbb{Z}_{100} \) have a greatest common divisor of 10 with 100. These elements are:
   \[
   a = 10, 20, 30, 40, 50, 60, 70, 80, 90
   \]
   since for each of these \(a\), \( \gcd(a, 100) = 10\).

### Conclusion:
The elements of order 10 in \( \mathbb{Z}_{100} \) are:
\[
\{10, 20, 30, 40, 50, 60, 70, 80, 90\}
\]

Would you like further details or have any questions?

### Related Questions:
1. What are the elements of order 5 in \( \mathbb{Z}_{100} \)?
2. How do we find elements of order 20 in \( \mathbb{Z}_{100} \)?
3. Can you explain how to determine the order of an element in any group?
4. What is the significance of the greatest common divisor in determining element order?
5. How many elements of order 25 are there in \( \mathbb{Z}_{100} \)?

### Tip:
When finding the order of an element in modular arithmetic, always start by determining the greatest common divisor (gcd) between the element and the modulus.

Discover how to find elements of order 10 in Z100 using modular arithmetic and the greatest common divisor. This step-by-step guide covers group theory concepts and key formulas for understanding element orders. Ideal for undergraduate students studying abstract algebra.

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