The notation 〈6〉 in \( \mathbb{Z}_{16} \) refers to the cyclic subgroup generated by 6 under addition modulo 16. To find the elements of this cyclic subgroup, we repeatedly add 6 modulo 16 until the sequence repeats.

Starting with 6, we have:

1. \( 6 \mod 16 = 6 \)
2. \( 6 + 6 = 12 \mod 16 = 12 \)
3. \( 12 + 6 = 18 \mod 16 = 2 \)
4. \( 2 + 6 = 8 \mod 16 = 8 \)
5. \( 8 + 6 = 14 \mod 16 = 14 \)
6. \( 14 + 6 = 20 \mod 16 = 4 \)
7. \( 4 + 6 = 10 \mod 16 = 10 \)
8. \( 10 + 6 = 16 \mod 16 = 0 \)

Thus, the elements of 〈6〉 in \( \mathbb{Z}_{16} \) are:
\[
\{0, 2, 4, 6, 8, 10, 12, 14\}
\]

These are the distinct elements of the cyclic subgroup generated by 6 in \( \mathbb{Z}_{16} \).

Would you like more details or have any questions?  
Here are some related questions to explore further:

1. What are the elements of 〈2〉 in \( \mathbb{Z}_{16} \)?
2. What is the order of the cyclic group 〈6〉 in \( \mathbb{Z}_{16} \)?
3. How do we determine the generator of a cyclic group in \( \mathbb{Z}_n \)?
4. What is the greatest common divisor of 6 and 16, and why is it relevant here?
5. How does the structure of cyclic groups change with different moduli?

**Tip:** The order of an element \( a \) in \( \mathbb{Z}_n \) is the smallest integer \( k \) such that \( k \cdot a \equiv 0 \mod n \).

Explore the elements of the cyclic subgroup 〈6〉 in Z16, generated by 6 under addition modulo 16. Learn about modular arithmetic, cyclic groups, and how to find the elements in this cyclic group step-by-step. Suitable for advanced high school math students in Grades 11-12.

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